Welcome to Section 8-4! We have some excellent news to start the week: there will be no quiz this week. However, this is not a week to slack off; instead, please use this extra time to work on your final presentations. Make them interesting, engaging, and mathematical!
Section 8-4: Apportionment - Am I Represented?
In this section, we tackle the complex problem of Apportionment. This is the method used to determine the ideal district size for a House representative and how many representatives each state gets based on its population.
The fundamental challenge is that when you divide populations, you rarely get whole numbers, but you cannot have a fraction of a representative. We must decide how to handle these fractional parts fairly.
Key Concepts
- Standard Divisor (Ideal District Size): This represents the average number of people represented by one seat.
$$ \text{Ideal District Size} = \frac{\text{Total U.S. Population}}{\text{House Size}} $$ - Standard Quota: This is the exact share of seats a state should get if decimals were allowed.
$$ \text{State's Quota} = \frac{\text{State's Population}}{\text{Ideal District Size}} $$
Evolution of Apportionment Methods
Historically, the U.S. has used several methods to resolve the issue of fractional quotas. Here is a breakdown of the methods covered in the notes:
- Hamilton's Method (Largest Remainder):
Calculate the quota and round down to the nearest whole number. If there are leftover seats, give them to the states with the largest fractional remainders.
Note: This method is susceptible to the Alabama Paradox, where increasing the total number of house seats can actually cause a state to lose a representative. - Jefferson's Method (Adjusted Divisor):
Round all quotas down. If the total number of seats is too low, we adjust the divisor down to increase the quotas. - Adams's Method (Adjusted Divisor):
Round all quotas up. If the total is too high, we adjust the divisor up. - Webster's Method (Adjusted Divisor):
Round the quota to the nearest integer (standard rounding). Adjust the divisor as necessary until the seats add up correctly.
The Modern Solution: Huntington-Hill Method
Currently, the United States uses the Huntington-Hill Method. This is an adjusted divisor method that uses the Geometric Mean to decide whether to round up or down.
The Geometric Mean of a number $n$ and $n+1$ is calculated as:
$$ \text{Geometric Mean} = \sqrt{n(n+1)} $$
If a state's quota is larger than the geometric mean of its lower and upper bounds, we round up; otherwise, we round down.
Review the attached slides to see detailed examples of these calculations for states like Montana, Alaska, and Florida. Good luck with your presentations!