Procedure:

1. Calculate the Standard Divisor.
2. Calculate each state’s Standard Quota.
3. Initially assign each state its Lower Quota.
4. If there are surplus seats, give them, one at a time, to states in descending order of the fractional parts of their Standard Quota.

Problems:

• The Alabama Paradox
  An increase in the total number of seats to be apportioned causes a state to lose a seat.
• The Population Paradox
  An increase in a state’s population can cause it to lose a seat.
• The New States Paradox
  Adding a new state with its fair share of seats can affect the number of seats due other states.

Procedure:

1. Calculate the Standard Divisor.
2. Calculate each state’s Standard Quota.
3. Initially assign each state its Lower Quota.
4. Check to see if the sum of the Lower Quotas is equal to the correct number of seats to be apportioned.
   • If the sum of the Lower Quotas is equal to the correct number of seats to be apportioned, then apportion to each state the number of seats equal to its Lower Quota.
   • If the sum of the Lower Quotas is NOT equal to the correct number of seats to be apportioned, then, by trial and error, find a number, MD, called the Modified Divisor to use in place of the Standard Divisor so that when the Modified Quota, MQ, for each state (computed by dividing each State's Population by MD instead of SD) is rounded DOWN, the sum of all the rounded (down) Modified Quotas is the exact number of seats to be apportioned. (Note: The MD will always be smaller than the Standard Divisor.) These rounded (down) Modified Quotas are sometimes called Modified Lower Quotas. Apportion each state its Modified Lower Quota.

Problem:

• Violates the Quota Rule. (However, it can only violate Upper Quota—never Lower Quota.)

Procedure:

1. Calculate the Standard Divisor.
2. Calculate each state’s Standard Quota.
3. Initially assign each state its Upper Quota.
4. Check to see if the sum of the Upper Quotas is equal to the correct number of seats to be apportioned.
   • If the sum of the Upper Quotas is equal to the correct number of seats to be apportioned, then apportion to each state the number of seats equal to its Upper Quota.
   • If the sum of the Upper Quotas is NOT equal to the correct number of seats to be apportioned, then, by trial and error, find a number, MD, called the Modified Divisor to use in place of the Standard Divisor so that when the Modified Quota, MQ, for each state (computed by dividing each State's Population by MD instead of SD) is rounded UP, the sum of all the rounded (up) Modified Quotas is the exact number of seats to be apportioned. (Note: The MD will always be larger than the Standard Divisor.) These rounded (up) Modified Quotas are sometimes called Modified Upper Quotas. Apportion each state its Modified Upper Quota.

Problem:

• Violates the Quota Rule. (However, it can only violate Lower Quota—never Upper Quota.)

Procedure:

1. Calculate the Standard Divisor.
2. Calculate each state’s Standard Quota.
3. Initially assign a state its Lower Quota if the fractional part of its Standard Quota is less than 0.5.
   Initially assign a state its Upper Quota if the fractional part of its Standard Quota is greater than or equal to 0.5.
   [In other words, round down or up based on the arithmetic mean (average).]
4. Check to see if the sum of the Quotas (Lower and/or Upper from Step 3) is equal to the correct number of seats to be apportioned.
   • If the sum of the Quotas (Lower and/or Upper from Step 3) is equal to the correct number of seats to be apportioned, then apportion to each state the number of seats equal to its Quota (Lower or Upper from Step 3).
   • If the sum of the Quotas (Lower and/or Upper from Step 3) is NOT equal to the correct number of seats to be apportioned, then, by trial and error, find a number, MD, called the Modified Divisor to use in place of the Standard Divisor so that when the Modified Quota, MQ, for each state (computed by dividing each State's Population by MD instead of SD) is rounded based on the arithmetic mean (average) , the sum of all the rounded Modified Quotas is the exact number of seats to be apportioned. Apportion each state its Modified Rounded Quota.

Problem:

• Violates the Quota Rule. (However, violations are rare and are usually associated with contrived situations.)

• Current method used to apportion U.S. House
• Developed around 1911 by Joseph A. Hill, Chief Statistician of the Bureau of the Census and Edward V. Huntington, Professor of Mechanics & Mathematics, Harvard
• Preliminary terminology: The Geometric Mean

Procedure:

1. Calculate the Standard Divisor.
2. Calculate each state’s Standard Quota.
3. Initially assign a state its Lower Quota if the fractional part of its Standard Quota is less than the Geometric Mean of the two whole numbers that the Standard Quota is immediately between (for example, 16.47 is immediately between 16 and 17).
   Initially assign a state its Upper Quota if the fractional part of its Standard Quota is greater than or equal to the Geometric Mean of the two whole numbers that the Standard Quota is immediately between (for example, 16.47 is immediately between 16 and 17).
   [In other words, round down or up based on the geometric mean.]
4. Check to see if the sum of the Quotas (Lower and/or Upper from Step 3) is equal to the correct number of seats to be apportioned.
   • If the sum of the Quotas (Lower and/or Upper from Step 3) is equal to the correct number of seats to be apportioned, then apportion to each state the number of seats equal to its Quota (Lower or Upper from Step 3).
   • If the sum of the Quotas (Lower and/or Upper from Step 3) is NOT equal to the correct number of seats to be apportioned, then, by trial and error, find a number, MD, called the Modified Divisor to use in place of the Standard Divisor so that when the Modified Quota, MQ, for each state (computed by dividing each State's Population by MD instead of SD) is rounded based on the geometric mean, the sum of all the rounded Modified Quotas is the exact number of seats to be apportioned. Apportion each state its Modified Rounded Quota.

Problem:

• Violates the Quota Rule.