	The House of Representatives Apportionment Formula: An Analysis of Proposals for Change and Their Impact on States1

		Introduction

As a basis for understanding the reallocation of Representatives among the states based on the 2010 Census, it may prove helpful to examine the current House of Representatives apportionment formula. In addition, some members of the statistical community have, in the recent past, urged Congress to consider changing the current apportionment formula. Consequently, an examination of other methods that could be used to apportion the seats in the House of Representatives may contribute to a deeper understanding of the apportionment process. One, the Hamilton-Vinton method, involves ranking fractional remainders. One solution to this problem of too few or too many seats would be to divide each state's population by the national "ideal" size district, but instead of rounding at the .5 point, allot each state initially the whole number of seats in its quota (except that states entitled to less than one seat would receive one regardless because of constitutional requirements). Next, rank the fractional remainders of the quotas in order from largest to smallest. If the number of Representatives assigned using the whole numbers is less than the House total, rank the fractional remainders of the states' quotas and award seats in rank order from highest to lowest until the House size is reached. In the Adams, Webster, and Jefferson methods, the rounding points used are the same for a state of any size. In the Dean and Hill methods, on the other hand, the rounding point varies with the number of seats assigned to the state; it rises as the state's population increases. These differences among the rounding methods are illustrated in Figure 1 . Hill: Rounding at the Geometric Mean

The only operational difference between a Webster and a Hill apportionment (equal proportions—the method in use since 1941), is where the rounding occurs. The Dean method (which has never been used) can be defined in the following manner for a 435-seat House:
Dean
Find a number so that when it is divided into each state's population and resulting quotients are rounded at the harmonic mean, the total number of seats will sum to 435. (In all cases where a state would be entitled to less than one seat, it receives one anyway because of the constitutional requirement.) Each apportionment method discussed in this report has a "rational" basis, and for each, there is at least one test according to which it is the most equitable. The question of how the concept of fairness can best be defined, in the context of evaluating an apportionment formula, remains open. Some rounding methods are better than others in this respect. Which of the mathematical tests discussed in this report best approximates the constitutional requirement that Representatives be apportioned among the states according to their respective number is, arguably, a matter of judgment—rather than an indisputable mathematical test.
As a basis for understanding the reallocation of Representatives among the states based on the 2010 Census, it may prove helpful to examine the current House of Representatives apportionment formula. In addition, some members of the statistical community have, in the recent past, urged Congress to consider changing the current apportionment formula. Consequently, an examination of other methods that could be used to apportion the seats in the House of Representatives may contribute to a deeper understanding of the apportionment process.
Seats in the House of Representatives are allocated by a formula known as the method of equal proportions or the "Hill" method. If Congress decided to change it, there are at least five alternatives it might consider. Four of these are based on rounding fractions and one, on ranking fractions. The current apportionment system (codified in 2 U.S.C. 2a) also is based on rounding fractions.
The Hamilton-Vinton method, used to apportion the House of Representatives from 1851-1901, is based on ranking fractions. First, the total population of the 50 states is divided by 435 (the House size) in order to find the national "ideal sized" district. Next, this number is divided into each state's population, producing the state's "quota" of seats. Each state is then awarded the whole number in its quota (but at least one). If fewer than 435 seats have been assigned by this process, the fractional remainders of the 50 states are rank-ordered from largest to smallest, and seats are assigned in this manner until 435 are allocated.
The rounding methods, including the Hill method currently in use, allocate seats among the states differently, but operationally the methods only differ by where rounding occurs in seat assignments. Three of these methods—Adams, Webster, and Jefferson—have fixed rounding points. Two others—Dean and Hill—use varying rounding points that rise as the number of seats assigned to a state grows larger. The methods can be defined in the same way (after substituting the appropriate rounding principle in parentheses). The rounding point for Adams is (up for all fractions); for Dean (at the harmonic mean); for Hill (at the geometric mean); for Webster (at the arithmetic mean, which is 0.5 for successive numbers); and for Jefferson (down for all fractions). Substitute these phrases in the general definition below for the rounding methods:
Find a number so that when it is divided into each state's population and resulting quotients are rounded (substitute appropriate phrase), the total number of seats will sum to 435. (In all cases where a state would be entitled to less than one seat, it receives one anyway because of the constitutional requirement.)
Fundamental to choosing an apportionment method is a determination of fairness. Each apportionment method discussed in this report has a "rational" basis, and for each, there is at least one test according to which it is the most equitable. The question of how the concept of fairness can best be defined, in the context of evaluating an apportionment formula, remains open. Which of the mathematical tests discussed in this report best approximates the constitutional requirement that Representatives be apportioned among the states according to their respective numbers is, arguably, a matter of judgment, rather than an indisputable mathematical test.