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Apportionment

Methods of Apportionment

Following each decennial census, the seats of the House of Representatives are reapportioned among the states according to their population. There are several different methods of apportionment, listed below, each of which allocates seats in a slightly different way.

The Hamilton/Vinton Method

The Hamilton/Vinton Method sets the divisor as the proportion of the total population per house seat. After each state's population is divided by the divisor, the whole number of the quotient is kept and the fraction dropped. This will result in surplus house seats. The first surplus seat is assigned to the state with the largest fraction after the original division. The next is assigned to the state with the second-largest fraction and so on.For example:

If a country had 4 states, and a 20-seat House of Representatives...

2560 + 3315 + 995 + 5012 = 11882
11882 ⁄ 20 = 594.1

State Population Quotient First Allocation of Seats Left Over Decimal Seats Apportioned
A 2560 2560 ⁄ 594.1 = 4.31 4 .31 4
B 3315 3315 ⁄ 594.1 = 5.58 5 .58 6
C 995 995 ⁄ 594.1 = 1.67 1 .67 2
D 5012 5012 ⁄ 594.1 = 8.44 8 .44 8
Total = 20 Seats

The Jefferson Method

The Jefferson Method avoids the problem of an apportionment resulting in a surplus or a deficit of House seats by using a divisor that will result in the correct number of seats being apportioned. For example:

If a country had 4 states, and a 20-seat House of Representatives...

2560 + 3315 + 995 + 5012 = 11882
11882 ⁄ 20 = 594.1

State Population Quotient Seats Apportioned
A 2560 2560 ⁄ 594.1 = 4.31 4
B 3315 3315 ⁄ 594.1 = 5.58 5
C 995 995 ⁄ 594.1 = 1.67 1
D 5012 5012 ⁄ 594.1 = 8.44 8
Total = 18 Seats (2 Surplus)

But if the divisor were 550 instead of 594.a...

State Population Quotient Seats Apportioned
A 2560 2560 ⁄ 550 = 4.65 4
B 3315 3315 ⁄ 550 = 6.03 6
C 995 995 ⁄ 550 = 1.81 1
D 5012 5012 ⁄ 550 = 9.11 9
Total = 20 Seats

The Webster Method

The Webster Method is a modified version of the Hamilton/Vinton method. After the state populations are divided by the divisor, those with quotients that have fractions of 0.5 or above are awarded an extra seat. States with a quotient with a fraction below 0.5 have the fraction dropped. The size of the house of representatives is set in order to calculate the divisor, but can be increased in the final apportionment if a large number of states have fractions above 0.5.

If a country had 4 states, and a planned 20-seat House of Representatives...

2560 + 3315 + 995 + 5012 = 11882
11882 ⁄ 20 = 594.1

State Population Quotient First Allocation of Seats Left Over Decimal Seats Apportioned
A 2560 2560 ⁄ 594.1 = 4.31 4 .31 4
B 3315 3315 ⁄ 594.1 = 5.58 5 .58 6
C 995 995 ⁄ 594.1 = 1.67 1 .67 2
D 5012 5012 ⁄ 594.1 = 8.44 8 .44 8
Total = 20 Seats

The Huntington-Hill Method

The Huntington-Hill Method is a modified version of the Webster method, but it uses a slightly different rounding method. While Webster's method rounds at 0.5, the Huntington-Hill method rounds at the geometric mean, which is described below. If a state's quotient is higher than its geometric mean, it will be allocated an additional seat. This method will almost always result in the desired number of seats.

The geometric mean of two numbers is the square root of their product.

For example, the arithmetic mean of 4 and 5 is 4.5:
(4 + 5) ⁄ 2 = 4.5

The geometric mean is 4.47:
(4 x 5) = 20
√(20) = 4.47

If a country had 4 states, and a planned 20-seat House of Representatives...

2560 + 3315 + 995 + 5012 = 11882
11882 ⁄ 20 = 594.1

State Population Quotient Lower Quotient Upper Quotient Geometric Mean Seats Apportioned
A 2560 2560 ⁄ 594.1 = 4.31 4 5 4.47 4
B 3315 3315 ⁄ 594.1 = 5.58 5 6 5.48 6
C 995 995 ⁄ 594.1 = 1.67 1 2 1.41 2
D 5012 5012 ⁄ 594.1 = 8.44 8 9 8.49 8
Total = 20 Seats

Source: U.S. Census Bureau | Census History Staff | Last Revised: March 31, 2014