Activity 14.2

Divisor Methods

10 points

Due (at my office) at 12:00 noon, Thursday, April 2, 2009

In this activity, you will work with two divisor methods of apportionment: Adam's method and the Huntington-Hill method. With the help of Excel, you will determine how many representatives in the House of Representatives would correspond to each state if these methods were used.

Each method uses a modified divisor, that is, a divisor other than the standard divisor, to determine each state's modified quota. If the divisor D is used, then a state's modified quota is:

modified quota = state's population / D

Then, depending on the method used, all modified quotas are rounded up or down. In Adam's method, they are all rounded up; in Jefferson's method, they are rounded down. In Webster's method each modified quota is rounded up or down to its nearest integer. The method currently used in the U.S. to apportion representatives is the Huntington-Hill method. The method uses the geometric mean to round off quotas.

Adam's Method
1. In this method modified quotas are rounded up; thus the modified divisor needs to be greater than the standard divisor. Why? Explain what would happen if the modified divisor were less than the standard divisor.

2. Open the file EA14.1 State Population.xls . Compute the total U.S. population in cell B53. Name this cell using the Name box above column A on the spreadsheet. (You could use the name total, or any other appropriate name). Also, in cell A53, type the text label, Total Population = . Enter the total population in your Word document.

3. Compute the standard divisor in cell B55 using the relationship:
standard divisor = total population / total number of seats
Remember that the total number of seats remains 435.

Record this number in your Word document. In cell A55, enter the text Standard divisor = .

4. In column C, enter each state's standard quota using the relationship:

standard quota = state's population / standard divisor

Use cell C53 to find the sum of all states' standard quotas and enter this sum in your Word document. Label column C by entering Standard Quota in cell C1.

5. In cell A56, enter the text Divisor = and in cell D56 enter a number greater than the standard divisor. (Pick any number greater than the standard divisor - you'll change it later to find the "best" one.) Be sure to enter this new divisor in your Word document.

6. Label column D by entering Modified Quota in D1. In cell D2, enter an appropriate formula and then drag it down to compute the modified quota for each state obtained by using the modified quota for each state obtained by using the modified divisor you entered in cell D56. (Make sure you enter a formula that will keep the location "D56" fixed when you drag the formula. Recall that you can do this by either naming cell D56 or by using $ signs.)

7. In each cell of column E, enter the integer number obtained by rounding up the modified quota. (To accomplish this enter =INT(D2) + 1 in cell E2 and drag to autofill the column.) Label the column Number of Seats Using Adam's Method.

8. In cell E53, enter the sum of the numbers in cells E2 through cell E51. Enter this sum in your Word document, and explain what this number means.

9. If the previous sum is exactly 435, you have succeeded in using Adams' method to apportion House seats. Explain why this is so.

10. If the sum is not 435, then you need to try other numbers for the modified divisor until 435 is reached. Note that when you change the number in cell D56 (the modified divisor), the numbers in column E, including the sum in E53, will also change automatically.

Using the table below, record the divisors you've tried and the total number of House seats resulting from each divisor. Indicate which one is the modified divisor that actually works. Include this table in your Word document.

11. List the four states that would get the largest number of seats if you used Adam's method and the number of seats each would get.

12. List the four states that would get the smallest number of seats if you used Adam's method and the number of seats each would get.

Huntington-Hill Method
In this method, each modified quota is rounded up or down according to the geometric mean. The geometric mean of two numbers x and y is the square root of the product of the two numbers. That is,

Geometric mean of x and y = square root of ( x * y )

13. Find the geometric mean of 33 and 47, and compare it with the mean average of the same two numbers.

14. In the Huntington-Hill method, the geometric mean of the integer part of the modified quota and the next integer is calculated. If the modified quota is greater than the geometric mean, then the modified quota is rounded up. Otherwise, it is rounded down. For example, if the modified quota is 12.742, the geometric mean of 12 and 13 is calculated square root of ( 12 * 13 ) = 12.489. Because the modified quota 12.742 is greater than 12.489 (the geometric mean of 12 and 13), the modified quota is rounded up to 13.

Suppose the modified quota is 9.49. Would the modified quota be rounded up or down in this method? Show your computations and explain your conclusion.

15. The following problem is based on Exploration #3, of Topic 14, on page 253 in your textbook. You do NOT need to perform your calculations in Excel, but you need to show your work in your Word document.

Three colleges in the same region, Colleges A, B, and C, decide to form a ten-member committee of students representing the three colleges. It is decided that representation will be proportional to the number of students in each college in such a way that each college will have at least one representative. College A has 1,455 students, College B has 1,683 students, and College C has 2,706 students.

15. a. Find the standard divisor and each college's standard quota.

15. b. Use Hamilton's method to apportion the number of committee members to each college.

15. c. Use Lowndes' method to apportion the number of committee members to each college.

15. d. Which of the two methods would the students of College A prefer? Briefly explain why.

15. e. If we use the Huntington-Hill method with a modified divisor of 750, how many members would each college have?

15. f. Is 750 a correct choice of modified divisor to apportion the ID slots using the Huntington-Hill method? Would you increase it or decrease it?

Summary
In this activity, you used Excel to investigate two divisor methods of apportionment: Adam's Method and the Huntington-Hill method, which is the method currently used to apportion seats in the House of Representatives. Using each of these methods, you found the number of representatives that corresponds to each state. You learned how to use Excel to compute the geometric mean and to round numbers according to the geometric mean.