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

Royce Crocker
Specialist in American National Government

In preparation for 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 Hill," or equal proportions, method. If Congress decided to change it, there are at least five alternatives to consider. Four of these are based on rounding fractions and one, on ranking fractions. The current apportionment system (codified in 2 U.S.C. 2a) is one of the rounding methods. 

The Hamilton-Vinton method is based on ranking fractions. First, the population of 50 states is divided by 435 (the House size) in order to find the national "ideal size" district. Next, this number is divided into each state's population. Each state is then awarded the whole number in its quotient (but at least one). If fewer than 435 seats have been assigned by this process, the fractional remainders of the 50 states are rank-ordered from largest to smallest, and seats are assigned in this manner until 435 are allocated. 

The rounding methods, including the Hill method currently in use, allocate seats among the states differently, but operationally the methods only differ by where rounding occurs in seat assignments. Three of these methods—Adams, Webster, and Jefferson—have fixed rounding points. Two others—Dean and Hill—use varying rounding points that rise as the number of seats assigned to a state grows larger. The methods can be defined in the same way (after substituting the appropriate rounding principle in parentheses). The rounding point for Adams is (up for all fractions); for Dean (at the harmonic mean); for Hill (at the geometric mean); for Webster (at the arithmetic mean, 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.

Date of Report: August 26, 2010
Number of Pages: 28
Order Number: R41382
Price: $29.95

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