The U.S. House of Representatives Apportionment Formula in Theory and Practice
Released: 2013-08-02T00:00:00

Summary:
On December 21, 2010, the number of seats allocated to each state for the House of Representatives was announced. 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 2010 apportionment, this could have meant a House of Representatives as small as 50 or as large as 10,306 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.

subsections: section_title: The U.S. House of Representatives Apportionment Formula in Theory and Practice1
subsections: section_title: Introduction
paragraphs: 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. 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. 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.
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. 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.
section_title: Constitutional and Statutory Requirements
paragraphs: 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.
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 showing the whole number of persons in each state" and the resulting seat allocation within one week after the opening of the first regular session of Congress following the census.
The Census Bureau has been assigned the responsibility of computing the apportionment. As a matter of practice, the Director of the Bureau reports the results of the apportionment at the end of December 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.
section_title: The Apportionment Formula
subsections: section_title: The Formula in Theory
paragraphs: An intuitive way to apportion the House is through simple rounding (a method never adopted by Congress). First, the U.S. apportionment population is divided by the total number of seats in the House (e.g., 309,183,463 divided by 435, in 2010) to identify the "ideal" sized congressional district (708,377 in 2010). 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.
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. 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 2010 apportionment, the "ideal" size district of 708,377 had to be adjusted upward to between 709,063 and 710,231 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.
section_title: The Formula in Practice: Deriving the Apportionment from a Table of "Priority Values"
paragraphs: 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 20 th 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.
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 re-computations.
The traditional method of constructing a priority list to apportion seats by the equal proportions method involves first computing the reciprocals 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 27 seats must be apportioned using the equal proportions formula. The 2010 apportionment populations for these states were 37,341,989 for California, 19,421,055 for New York, and 18,900,773 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. 
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 26,404,773.64 (0.70710681 x 37,341,989), while New York's claim to a second seat is 13,732,759.69 (0.70710681 x 19,421,055), and Florida's claim to a second seat is 13,364,864.76 (0.70710681 x 18,900,773). Based on the priority values, California has the highest claim for its second seat and is allocated the fourth seat in the hypothesized chamber. 
Next, the fifth seat's allocation is determined. California's claim to a third seat, based on the computed priority value, is 15,244,803.17 (0.40824829 x 37,341,989), while, as above, New York's claim to its second seat is 13,732,759.69 (0.70710681 x 19,421,055) and Florida's claim to its second seat is 13,364,864.76 (0.70710681 x 18,900,773). 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 10,779,703.70 (0.288675135 x 37,341,989), while, as above, New York's claim to its second seat is 13,732,759.69 (0.70710681 x 19,421,055) and Florida's claim to its second seat is 13,364,864.76 (0.70710681 x 18,900,773). 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 10,779,703.70 (0.288675135 x 37,341,989), while, having received its second seat, New York's claim to its third seat is 7,928,612.50(0.40824829 x 19,421,055) and Florida's claim to its second seat is 13,364,864.76 (0.70710681 x 18,900,773). 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 11 seats, New York would have 10, and Florida would have 9. Any other size House can be determined by picking points in the priority list and observing what the maximum size state delegation would be for each state.
A priority listing for all 50 states based on the 2010 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.
section_title: Challenges to the Current Formula
paragraphs: 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." 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."
A bill that would have changed the apportionment method to another formula called the "Hamilton-Vinton" method was introduced in 1981. The fundamental principle of the Hamilton-Vinton method is that it ranks fractional remainders. In order to reapportion the House using Hamilton-Vinton, each state's population would be divided by the "ideal" sized congressional district (309,183,463 divided by 435, in 2010, for an "ideal" district population of 708,377). 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.
section_title: Equal Proportions or Major Fractions: An Analysis
paragraphs: 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.
subsections: section_title: The Case for Major Fractions
paragraphs: 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 2010, the state of North Carolina would have been assigned 14 seats under the major fractions method, and the state of Rhode Island would have received 1 seat. Under this allocation, there would have been 1.4636 Representatives per million for North Carolina residents and 0.9476 Representatives per million for Rhode Island residents. The absolute value of the difference between these two numbers is 0.5160.
Under the equal proportion method of assigning seats in 2010, North Carolina actually received 13 seats and Rhode Island 2. With 13 seats, North Carolina received 1.3590 Representatives for each million persons, and Rhode Island, with 2 seats, received 1.8953 Representatives per million persons. The absolute value of the difference between these two numbers is 0.5363. 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 Karche r v. Daggett , 462 U.S.725 (1983).
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 that 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). 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, (3) the number of states, and (4) the method of apportionment. A 1984 exercise prepared for the Congressional Research Service (CRS) involving 1,000 simulated apportionments examined the results when two of these variables were changed—the method and the state populations. In order to further approximate reality, the state populations used in the apportionments were based on the Census Bureau's 1990 population projections available at that time. Each method was tested by computing 1,000 apportionments and tabulating the results by state. There was no discernible pattern by size of state in the results of the major fractions apportionment. The equal proportions exercise, however, showed that the smaller states were persistently advantaged.
Another way of evaluating the impact of a possible change in apportionment methods is to determine the odds of an outcome being different than the one produced by the current method—equal proportions. If equal proportions favors small states at the expense of large states, would switching to major fractions, a method that appears not to be influenced by the size of a state, increase the odds of the large states gaining additional representation? Based on the simulation model prepared for CRS, this appears to be true. The odds of any of the 23 largest states gaining an additional seat in any given apportionment range from a maximum of 13.4% of the time (California) to a low of .2% of the time (Alabama). The odds of any of the 21 multi-districted smaller states losing a seat range from a high of 17% (Montana, which then had two seats) to a low of 0% (Colorado), if major fractions were used instead of equal proportions.
In the aggregate, switching from equal proportions to major fractions "could be expected to shift zero seats about 37% of the time, to shift 1 seat about 49% of the time, 2 seats 12% of the time, and 3 seats 2% of the time (and 4 or more seats almost never), and, these shifts will always be from smaller states to larger states."
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.
section_title: The Case for Equal Proportions, the Current Method
paragraphs: 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.
An example using the North Carolina and Rhode Island 2010 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. The equal proportions formula assigned 13 seats to North Carolina and 2 to Rhode Island in 2010. If the major fractions method had been used, then 14 seats would have been assigned to North Carolina, and 1 would have been given to Rhode Island. Under this scenario, North Carolina has 1.4636 Representatives per million persons and Rhode Island has 0.9476 Representatives per million. The absolute difference between these numbers is 0.5160 and the proportional difference between the two states' Representatives per million is 54.45%. When 13 seats are assigned to North Carolina and 2 are assigned to Rhode Island (using equal proportions), North Carolina has 1.3590 Representatives per million and Rhode Island has 1.8953 Representatives per million. The absolute difference between these numbers is .0.5363 and the proportional difference is 39.46%.
Major fractions minimizes absolute differences, so in 2010, if this method had been required by law, North Carolina and Rhode Island would have received 14 and 1 seats respectively because the absolute difference (0.5160 Representatives per million) is smaller at 14 and 1 than it would be at 13 and 2 (0.5363). Equal proportions minimizes differences on a proportional basis, so it assigned 13 seats to North Carolina and 2 to Rhode Island because the proportional difference between a 13 and 2 allocation (39.46%) is smaller than would occur with a 14 and 1 assignment (54.45%).
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.
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 that are, perhaps, appropriate to consider? One such goal might be the desirability of avoiding large districts, if possible. After the apportionment of 2010, five of the seven states with only one Representative (Alaska, Delaware, Montana, North Dakota, South Dakota, Vermont, and Wyoming) have relatively large land areas. The five Representatives of the larger states will serve 1.22% of the U.S. population, but also will represent 27% of the U.S. total land area.
Arguably, an apportionment method that would potentially reduce the number of very large (with respect to area size) 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.
The table that follows contains the priority listing used in apportionment following the 2010 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.
subsections: section_title: Appendix. 2010 Priority List for Apportioning Seats to the House of Representatives