1. Section 15.3 Apportionment Methods

2. What You Will Learn Standard Divisor Standard Quota Lower Quota
Upper Quota
Hamilton’s Method
The Quota Rule
Jefferson’s Method
Webster’s Method
Adam’s Method

3. Apportionment
The goal of apportionment is to determine a method to allocate the total number of items to be apportioned in a fair manner.

4. Apportionment Four Methods Hamilton’s method Jefferson’s method
Webster’s method
Adams’s method

5. Standard Divisor
To obtain the standard divisor when determining apportionment, use the following formula.

6. Standard Quota
To obtain the standard quota when determining apportionment, use the following formula.

7. Example 1: Determining Standard Quotas
The Shanahan Law Firm needs to apportion 60 new fax machines to be distributed among the firm’s five offices. Since the offices do not all have the same number of employees, the firm’s managing partner decides to apportion the fax machines based on the number of employees at each office.

8. Example 1: Determining Standard Quotas
Determine the standard quotas for offices B, C, D, and E of the Shanahan Law Firm and complete the table. Use 18 as the standard divisor, as was determined above.

9. Example 1: Determining Standard Quotas
Solution
Office B: 201÷18 = 11.17, rounded.
Other offices’ standard quotas found in a similar manner.

10. Lower and Upper Quota
The lower quota is the standard quota rounded down to the nearest integer.
The upper quota is the standard quota rounded up to the nearest integer.

11. Hamilton’s Method
To use Hamilton’s method for apportionment, do the following.
1. Calculate the standard divisor for the set of data.
2. Calculate each group’s standard quota.

12. Hamilton’s Method
3. Round each standard quota down to the nearest integer (the lower quota). Initially, each group receives its lower quota.
4. Distribute any leftover items to the groups with the largest fractional parts until all items are distributed.

13. Example 2: Using Hamilton’s Method for Apportioning Fax Machines
Use Hamilton’s method to distribute the 60 fax machines for the Shanahan Law Firm discussed in Example 1.

14. Example 2: Using Hamilton’s Method for Apportioning Fax Machines
Solution

15. Example 2: Using Hamilton’s Method for Apportioning Fax Machines
Solution
Sum of the lower quotas is 57, leaving 3 additional fax machines to distribute.
Offices C, D, and A have the three highest fractional parts (0.89, 0.72, and 0.67) in the standard quota, each receives one of the additional fax machines using Hamilton’s method.

16. The Quota Rule
An apportionment for every group under consideration should always be either the upper quota or the lower quota.

17. Jefferson’s Method
1. Determine a modified divisor, d, such that when each group’s modified quota is rounded down to the nearest integer, the total of the integers is the exact number of items to be apportioned. We will refer to the modified quotas that are rounded down as modified lower quotas.
2. Apportion to each group its modified lower quota.

18. Example 4: Using Jefferson’s Method for Apportioning Legislative Seats
The Republic of Geranium needs to apportion 250 seats in the legislature. Suppose that the population is 8,800,000 and that there are five states, A, B, C, D, and E. The 250 seats are to be divided among the five states according to their respective populations, given in the table.

19. Example 4: Using Jefferson’s Method for Apportioning Legislative Seats
Use Jefferson’s method to apportion the 250 legislature seats among the five states. The standard divisor is calculated to be 35,200.

20. Example 4: Using Jefferson’s Method for Apportioning Legislative Seats
Solution
With Jefferson’s method, the modified quota for each group needs to be slightly greater than the standard quota. To accomplish this we use a modified divisor, which is slightly less than the standard divisor.

21. Example 4: Using Jefferson’s Method for Apportioning Legislative Seats
Solution
Try 35,000 as a modified divisor, d.
State A: 1,003,200 ÷ 35,000 ≈ 28.66
The modified lower quota is 28.
Find other states quotas in a similar manner.
The table showing the apportionment is on the next slide.

22. Example 4: Using Jefferson’s Method for Apportioning Legislative Seats
Solution

23. Example 4: Using Jefferson’s Method for Apportioning Legislative Seats
Solution
Sum of modified lower quotas is 249.
It’s less than the 250 seats.
Since it is too low, we need to try a lower modified divisor.
Try 34,900.
State A: 1,003,200 ÷ 34,900 ≈ 28.74
Find the other quotas to get the table:

24. Example 4: Using Jefferson’s Method for Apportioning Legislative Seats
Solution

25. Example 4: Using Jefferson’s Method for Apportioning Legislative Seats
Solution
The sum of the modified lower quotas is now 250, our desired sum.
Each state is awarded the number of legislative seats listed in the table under the category of modified lower quota.

26. Webster’s Method
1. Determine a modified divisor, d, such that when each group’s modified quota is rounded to the nearest integer, the total of the integers is the exact number of items to be apportioned. We will refer to the modified quotas that are rounded to the nearest integer as modified rounded quotas.
2. Apportion to each group its modified rounded quota.

27. Example 5: Using Webster’s Method for Apportioning Legislative Seats
Consider the Republic of Geranium and apportion the 250 seats among the five states using Webster’s method.

28. Example 5: Using Webster’s Method for Apportioning Legislative Seats
Solution
From Example 3, round standard quotas to nearest integer, sum is:
= 251.
Sum is too high, use larger divisor.

29. Example 5: Using Webster’s Method for Apportioning Legislative Seats
Solution
Try 35,250 as modified divisor, d.
State A: 1,003,200 ÷ 35,250 = 28.46
Find the quotas for the other states in a similar manner to get the table on the next slide.

30. Example 5: Using Webster’s Method for Apportioning Legislative Seats
Solution

31. Example 5: Using Webster’s Method for Apportioning Legislative Seats
Solution
Our sum of the modified rounded quotas is 250, as desired.
Therefore, each state is awarded the number of legislative seats listed in the table under the category of modified rounded quota.

32. Adams’s Method
1. Determine a modified divisor, d, such that when each group’s modified quota is rounded up to the nearest integer, the total of the integers is the exact number of items to be apportioned. We will refer to the modified quotas that are rounded up as modified upper quotas.
2. Apportion to each group its modified upper quota.

33. Example 6: Using Adams’s Method for Apportioning Legislative Seats
Consider the Republic of Geranium. Apportion the 250 seats among the five states using Adams’s method.

34. Example 6: Using Adams’s Method for Apportioning Legislative Seats
Solution
With Adams’s method, the modified quota needs to be slightly smaller than the standard quota.
Divide by a larger divisor. Try 35,400.
State A: 1,003,200 ÷ 35,400 ≈ 28.34
Find other states’ quotas, similarly to get the table on the next slide.

35. Example 6: Using Adams’s Method for Apportioning Legislative Seats
Solution

36. Example 6: Using Adams’s Method for Apportioning Legislative Seats
Solution
Using d = 35,400 and rounding up to the modified quotas, we have a sum of 250 seats, as desired.
Therefore, each state will be awarded the number of legislative seats listed in the table under the category of modified upper quota.

37. Apportionment Methods
Of the four methods we have discussed in this section, Hamilton’s method uses standard quotas. Jefferson’ s method, Webster’ s method, and Adams’ s method all make use of a modified quota and can all lead to violations of the quota rule.

38. Apportionment Methods