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<q>
The result of rounding a number q in the usual way: round down if the fractional part of q is less than 0.5, and round up otherwise; For example, <86/57> = 2; This notation is not standard and is used only in this text.
<<q>>
The result of rounding a number q the Hill-Huntington way.
Absolute difference of two numbers
The result of subtracting the smaller number from the larger.
Adjusted quota
The product of the state's quota and a multiplier; The purpose of adjusting the quotas is to correct a failure of the rounded quotas to sum to the house size.
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.
Apportionment method
A systematic way of computing solutions of apportionment problems.
Apportionment problem
To round a list of fractions to whole numbers in a way that preserves the sum of the original fractions.
Critical multiplier
An adjustment factor that, applied to a state's quota, is just enough to change that stateÕs tentative apportionment.
District population
A state's population divided by its apportionment.
Divisor method
One of many apportionment methods in which the apportionments are determined by multiplying the quotas for the states by a common factor to obtain adjusted quotas; The apportionments are calculated by rounding the adjusted quotas; Divisor methods differ in their rounding rules; The methods of Jefferson, Webster, and Hill-Huntington are divisor methods.
Geometric mean
For positive numbers A and B, the geometric mean is defined to be the square root of A times B.
Hamilton method
An apportionment method advocated by Alexander Hamilton; This method 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.
Hill-Huntington method
A divisor method, named for the statistician Joseph Hill and the mathematician Edward V. Huntington, that minimizes relative differences in both representative shares and district populations; It is based on the Hill-Huntington way of rounding.
Jefferson method
A divisor method invented by Thomas Jefferson, based on rounding all fractions down.
Lower quota
The result of rounding a state's quota down to a whole number.
Population paradox
If changes in population cause one state's apportionment to decrease and another's to decrease, although the first state's population increased and the second state's population had increased proportionally less or had actually decreased, the population paradox has occurred; This paradox is possible with all apportionment methods except divisor methods.
Quota
A state's quota is the number of seats it would receive if fractional seats could be awarded.
Quota condition
A requirement that in every situation each state's apportionment is equal to either its lower quota or its upper quota; The Hamilton method satisfies this requirement but divisor methods do not.
Relative difference
The relative difference between two positive numbers is obtained by subtracting the smaller number from the larger, and expressing the result as a percentage of the smaller number; Thus, the relative difference of 120 and 100 is 20%.
Representative share
A state's representative share is 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.
Tentative apportionment
When apportioning by use of a divisor method, a state's tentative apportionment is initially obtained by rounding its quota, using the rounding rule associated with the method; In the process of calculating the apportionment, a state's tentative apportionment changes as it acquires or loses seats.
Upper quota
The result of rounding a state's quota up to a whole number.
Webster method
A divisor method of apportionment invented by Representative Daniel Webster; It is based on rounding fractions the usual way, and minimizes differences of representative share between states.