Review Vocabulary

The result of rounding a number down; for example, . (p. 578)

The result of rounding a number up to the next whole number; for example, . (p. 578)

Absolute difference The result of subtracting a smaller number from a larger number. (p. 601)

Alabama paradox A state loses a representative solely because the size of the House is increased. This paradox is possible with the Hamilton method but not with divisor methods. (p. 582)

Apportionment method A systematic way of computing solutions of apportionment problems. (p. 573)

Apportionment problem Given a list of fractions whose sum is a whole number, to round each fraction in a way that preserves the original sum. (p. 573)

Apportionment quotient The result of dividing a state’s quota by the divisor when apportioning by a divisor method. The apportionment quotient is rounded to obtain the state’s apportionment, using the particular rounding rule associated with the chosen divisor method. (pp. p. 585 p. 586)

Dean method A divisor method of apportionment that minimizes differences in the district populations of the states. (p. 598)

d’Hondt method An apportionment method that is equivalent to the Jefferson method. It is typically used to apportion seats in parliaments to political parties in proportion to their votes. (p. 592)

District population A state’s population divided by its apportionment. (p. 575)

Divisor method One of many apportionment methods in which the apportionments are determined by dividing the population of each state by a common divisor to obtain apportionment quotients. The apportionments are calculated by rounding the apportionment quotients. Divisor methods differ in their rounding rules. The Jefferson, Webster, Dean, and Hill-Huntington methods are divisor methods. (p. 586)

Equitable apportionment An apportionment is equitable— by a specified measure of inequity—if its inequity cannot be reduced by transferring a seat from one state to another state. The measures of inequity that we have considered are absolute differences in representative share (p. 599), absolute differences in district population (p. 575), and percentage differences (p. 601) in either representative share or district population. (p. 600)

Geometric mean For numbers and , neither of which is negative, the geometric mean is defined to be . (p. 602)

Hamilton method An apportionment method that assigns to each state either its lower quota or its upper quota.

The states that receive their upper quotas are those whose quotas have the largest fractional parts. (p. 578)

Hare method The Hamilton method applied to apportion seats to political parties in a parliament. (p. 584)

Hill-Huntington method A divisor method that minimizes percentage differences in both representative shares and district populations. This method has been used to apportion seats in the U.S. House of Representatives since 1941. (p. 602)

Hill-Huntington rounding A number is rounded down if it is less than the geometric mean of and ; otherwise it is rounded up. (p. 603)

Jefferson method A divisor method based on rounding all fractions down. Thus, if is the apportionment quotient of state , the state’s apportionment is . (pp. p. 585 p. 586)

609

Lower quota The whole number part of a state’s quota . (p. 578)

New states paradox After an apportionment, a new state joins the Union. The house size is increased by the number of seats that the new state would be apportioned by the method in use. Yet when the house is reapportioned, two or more states that had participated in the original apportionment find that their apportionments have changed. (p. 585)

Percentage difference (between two positive numbers) Subtract the smaller number from the larger and express the result as a percentage of the smaller number. Thus, the percentage difference between 120 and 100 is 20%. (p. 601)

Population paradox A situation in which the house size is unchanged and the apportionment of one state, , decreases although its population has increased, while another state, , loses population and gains a seat. (p. 583)

Quota The quota is the quotient of a state’s population divided by the standard divisor . The quota is the number of seats a state would receive if fractional seats could be awarded. (p. 576)

Quota condition A requirement that an apportionment method should always assign to each state either its lower quota or its upper quota in every situation. The Hamilton method satisfies this condition, but no divisor method does. (p. 590)

Representative share The state’s apportionment divided by its population. It is intended to represent the amount of influence a citizen of that state would have on his or her representative. (p. 599)

Sainte-Laguë method A way of calculating an apportionment that leads to the same result as the Webster method. It is typically used when apportioning seats in parliament to political parties. (p. 596)

Standard divisor The ratio of the total population to the house size . In a congressional apportionment problem, the standard divisor represents the average district population. (p. 575)

Tentative apportionment For the Hamilton method, a state’s lower quota (p. 578); for a divisor method, the appropriately rounded apportionment quotient. (p. 579)

Upper quota The result of rounding a state’s quota up to a whole number . (p. 578)

Webster method A divisor method of apportionment that is based on rounding fractions to the nearest whole number. The Webster method minimizes the absolute differences of representative share between states. (p. 594)