Activity 14.1

Methods of Apportionment

10 points

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

In this activity, you will work with the two quota methods of apportionment: Hamilton's method and Lowndes' method. You will use each of these methods to determine how many representatives in the U.S. House would correspond to each state if the method were currently used.

All apportionment methods assign to each state a number of representatives based on the state's population. The file below contains the population of each state, as given by the 2007 census estimate.

1. Open the file EA14.1 State Population.xls and then find the total U.S. population. To do this in Excel, you want to add all the states' populations, which are the numbers in cells B2 through B51. To do this, enter
=SUM( B2 : B51)
in cell B53. Name this cell using the Name box above column A on the spreadsheet. (You could use the name totalpopulation, or any other appropriate name). Also, in cell A53, type the text label, Total Population = . Enter the total population in your Word document.

2. In cell A54, enter the label Total Seats = and in cell B54, enter the number 435; this is the total number of seats in the House of Representatives. Name this cell. You might want to call it totalseats, or another appropriate name. Recall that the standard divisor is given by:
standard divisor = total population / total number of seats

Ask Excel to find the standard divisor in cell B55 and name this cell. Add the label Standard Divisor = in cell A55.

Enter the standard divisor in your Word document.

3. Recall that a state's standard quota is defined by the equation:
standard quota = state's population / standard divisor

In column C, enter each state's standard quota. First, enter the column title Standard Quota in cell C1. Then ask Excel to do the computations for you by entering the appropriate formula in cell C2. Drag-and-fill to enter the rest of the values.

4. Can each state's standard quota be the apportioned number of representatives for that state? Why or why not?

In both quota methods, apportionment is done by rounding up or down each state's standard quota.
In Hamilton's method, each state is initially given the number of representatives equal to the integer value of the state's standard quota, unless this number is 0. For this case the initial and final apportioned number of seats for that state is 1 (so that every state is represented).

5. Suppose that three states, State A, State B, and State C, have standard quotas as given in the table below. Fill in the last column of the table for the initial number of representatives apportioned to those states when using Hamilton's method. Please copy and paste this table into your Word document.

After the initial apportionment is done using the intege value (or 1 when the integer value is 0), the remaining seats are assigned to the states with a standard quota larger than 1. The states with the largest fractional part of the standard quota are each given an extra seat.

6. Suppose the total number representatives that correspond to the three states is 7. Record in the table the number of representatives (seats) that correspond to each of the three states using Hamilton's method. Please copy and paste this table into your Word document.

You will now use Excel to determine the number of representatives apportioned to each of the 50 states under Hamilton's method.

7. In column D of your spreadsheet, you'll ask Excel to enter the initial number of seats for each state; that is, enter the integer part of the state's standard quota, unless this is 0. If the integer part of the standard quota is 0, then you'll enter 1. To do this, enter the integer value of the first state's standard quota in cell D2 by typing =INT(C2) then drag to autofill. If column D contains any 0's, go through and replace each 0 with a 1. Label column D with an appropriate label.

Use an appropriate formula to enter in column E the difference of each entry in column C minus the corresponding entry in column D. Label column E Fractional Part. In your Word document, type the names of the states for which the entries in column E are negative. Explain why these are negative. Please copy and paste the spreadsheet into your Word document also.

8. Add the numbers in column D (use the =sum( ) function in Excel) to decide how many seats have already been apportioned. Enter that number into your Word document.

Determine how many seats remain after the initial apportionment. Enter that number into your Word document.

This number of remaining seats is how many you will apportion using the fractional part of the standard quota.

9. To apportion the remaining seats using the fractional part of the standard quota, sort the data in columns A, B, C, D, and E by "Fractional Part" in descending order.

In column F, enter the final number of seats apportioned by Hamilton's method to each state. (Note: you do not need to enter those numbers one by one, you can use a formula. Enter =D2 + 1 in cell F2 and drag until you have apportioned all the remaining seats: then have Excel copy the numbers in column D for the remaining states.) Label column F Seats Using Hamilton's Method. Then sort the data (remember to highlight all columns) by "Seats Using Hamilton's Method," to help answer parts (a) and (b) of this question.

9. a. List the four states that get the most number of seats and say how many seats each of them gets.

9. b. What is the smallest number of seats assigned? Which states get the smallest number of seats?

Lowndes' method also uses the standard quota to apportion representatives. However, Lowndes' method uses the relative fractional part of the standard quota instead of the fractional part. The relative fractional part is used to determine which states' standard quotas will be rounded up to fill out the remaining seats.

The relative fractional part is the following quotient:
relative fractional part = fractional part / integer part

10. Assuming that State A and State D have the following standard quotas, record their relative fractional parts in the table, and copy and paste this table in your Word document.

To apportion the number of seats for each of the 50 states by Lowndes' method, you will first compute the relative fractional part of each state's standard quota.

11. Insert two columns to the LEFT of column F in your spreadsheet. In the new column F, recompute the integer part of the standard quota for each state. (You must recompute the integer part of the standard quota, since you'd modified column D following the directions in #7 above.)  In cell G2, compute the relative fractional part of the standard quota for the state in row 2. Drag-and-fill to compute the relative fractional part for each state, and label column G Relative Fractional Part.

12. Notice that you see #DIV/0! in several places in column G. Explain why this error message occurred and what it means.

13. First, sort the data by "Relative Fractional Part," in descending order (the #DIV/0! error messages should be at the top). Next, label column I, Seats Using Lowndes' Method and start filling this column by giving one seat to each state showing the error message. Explain why you need to give each of these states one representative.

14. Finally, apportion the rest of the seats, filling column I, using Lowndes' method. Remember that you will start with the integer values in column F, and then use the relative fractional part to determine which states get one additional seat. Once your apportionment is complete, you should answer the questions below. They might be easier to answer if you sort the data by the number of seats apportioned by Lowndes' method.
14. a. List the four states that get the most number of seats and say how many seats each of them gets.

14. b. What is the smallest number of seats assigned? Which states get the smallest number of seats?

Please copy and paste the resulting spreadsheet into your Word document. For formatting purposes, you only need to include three columns: State, Seats Using Hamilton's Method, and Seats Using Lowndes' Method.

15. Name the states that would prefer Hamilton's method and those that would prefer Lowndes' method. Explain why you chose these states.

16. Which states, those with a large population or those with a small population, are favored by Lowndes' method? Please give an explanation, appealing to the math involved, why this is the case.

Summary
In this activity, you used Excel to investigate two quota methods of apportionment and compard the results of using them to apportion the seats of the House of Representatives. You learned the Excel command that gives the integer part of a real number and used it to find the fractional parts.