1. Apportionment Methods
Section 14.3
Apportionment Methods

2. What You Will Learn
Upon completion of this section, you will be able to:
Solve apportionment problems using Hamilton’s method
Solve apportionment problems using Jefferson’s method
Solve apportionment problems using Webster’s method
Solve apportionment problems using Adams’ method
Solve apportionment problems using Hunting-Hill 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 Five Methods Hamilton’s method Jefferson’s method
Webster’s method
Adams’s method
Huntington-Hill 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
Determine the standard quotas for clinics:
B, C, D, and E of the First Physician Organization and complete the table below. Use 18 as the standard divisor.

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

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

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

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.
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.

12. Example 2: Using Hamilton’s Method for Apportioning Doctors
Use Hamilton’s method to distribute the 60 doctors to the First Physicians Organization clinics discussed in Example 1.

13. Example 2: Using Hamilton’s Method for Apportioning Doctors
Solution

14. Example 2: Using Hamilton’s Method for Apportioning Doctors
Solution
Sum of the lower quotas is 57, leaving 3 additional doctors to distribute.
Clinics 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 doctors using Hamilton’s method.

15. Jefferson’s Method
1. Calculate each group’s standard quota, then find the lower quota of each group.
2. 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.
3. Apportion to each group its modified lower quota.

16. 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.

17. 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.

18. 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.

19. 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.

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

21. 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:

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

23. 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.

24. Webster’s Method
1. Calculate each group’s standard quota, then find the rounded quota of each group.
2. 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.
3. Apportion to each group its modified rounded quota.

25. 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.

26. 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.

27. 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.

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

29. 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.

30. Adams Method
1. Calculate each group’s standard quota, then find the upper quota of each group.
2. 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.
3. Apportion to each group its modified upper quota.

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

32. Example 6: Using Adams Method for Apportioning Legislative Seats
Solution
With Adams 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.

33. Example 6: Using Adams Method for Apportioning Legislative Seats
Solution

34. Example 6: Using Adams 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.

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

36. Apportionment Methods Problems

37. History of Apportionment Methods used in the House of Representatives
Starting after the first Census taken in 1790 –
3 years to calculate the reapportionment
** Hamilton’s Method was approved in 1790** but Vetoed by George Washington
Jefferson’s Method was used until after issues in 1822 and 1832
Webster’s Method was used in and had similar flaws
Hamilton’s Method was used starting 1852 – Flaws were later found with Hamilton in the late 1800’s
Webster’s Method was used AGAIN in 1900
Huntington-Hill is the current Method used, starting in 1942
*** Adams Method was never used***

38. Huntington-Hill Method
Today, Congress uses the Huntington-Hill Method also known as The Method of Equal Proportions or The Hill Method to apportion seats in the House of Representatives.

39. Huntington-Hill Method
1. Find the Standard Quota 2. Find the Lower Quota and Upper Quota 3. Find the Geometric Mean of LQ and UQ GM = (LQ)(UQ) 4. Compare Standard Quota and the Geo. Mean IF SQ > GM use the use the Upper Quota IF SQ < GM use the use the Lower Quota 5. THEN….

40. Huntington-Hill Method
5. Add up the Apportionment 6. If the Apportionment equals the amount of items to apportion you are done. 7. If not find a modified divisor and repeat steps 1-4

41. Huntington-Hill Example
A Metropolitan Area Rapid Transit Service operates 6 bus routes (A, B, C, D, E, F) and 130 buses.
The buses are apportioned among the routes based on the average number of passengers in the table: The standard divisor is Apportion the buses under the Huntington-Hill method.
Route A B C D E F
Total
Average # of Passengers
45,300
31,070
20,490
14,160
10,260
8,720
130,000
Route A B C D E F
Total
Standard Quota
45.30
31.07
20.49
14.16
10.26
8.72 X
Geometric Mean
45∙46
45.497
31∙32
31.496
20∙21
20.494
14∙15
14.491
10∙11
10.488
8∙9
8.485
Huntington-Hill Apportionment 45 31 20 14 10 9
129
We are one short of the amount to apportion and have to use a Modified Divisor. Try using 998 as the Modified Divisor .

42. Huntington-Hill Example Continued
Using 998 you get the following table:
Route A B C D E F
Total
Modified Quota
45.391
31.132
20.531
14.188
10.281
8.7375 X
Geometric Mean
45∙46
45.497
31∙32
31.496
20∙21
20.494
14∙15
14.491
10∙11
10.488
8∙9
8.485
Huntington-Hill Apportionment 45 31 21 14 10 9
130