Rounding decimals or fractions to whole numbers might seem to be one of the most boring subjects ever.

Transcription

1 Apportionment

2 Rounding decimals or fractions to whole numbers might seem to be one of the most boring subjects ever. However, as we will see, the method used in rounding can be of great significance. Some method of rounding is required to determine how many members of the House of Representatives is given to each state. We will investigate different methods of rounding, each of which has been used to apportion the house. 2

3 Seats in the U.S. House of Representatives are divided, or apportioned, based on the populations of the states. Theoretically, the number of representatives a state receives is proportional to its fraction of the U.S. population. That is, if a state has 1% of the population, it should have 1% of the representatives. In practice, what happens is more complicated. 3

4 The 1791 apportionment of the House of Representatives gave it 105 members. These seats had to be divided among the 15 states according to the respective populations of the states. For example, New York should receive the number of seats according to the formula Population of New York Population of the U.S. x 105 4

5 The 1790 census listed the population of the U.S. at 3,615,920 and that of New York at 331,589. So, New York should have received 331,589 / 3,615,920 x 105 = 9.63 seats in the house. Of course, this is impossible, since the number of house members must be a whole number. 5

6 The apportionment problem is then how to round to whole numbers the values obtained when computing how many representatives each state should receive. Congress first considered a method proposed by Alexander Hamilton. This method was introduced in a bill, but was vetoed by President Washington. 6

7 Thomas Jefferson then developed a different method, which was then approved, and was used until Jefferson s method was replaced in 1842 by one developed by Daniel Webster. In 1852, Hamilton s method replaced Webster s, and was used until Webster s method was reintroduced, and used until 1940, when it was replaced by the Hill-Huntington method, which has been used ever since. 7

8 While we will discuss the apportionment problem in terms of apportioning the House of Representatives, the ideas we discuss work just as well for other situations. The quota is the exact amount that would be allocated to a state if a whole number was not required. The quota for a given state is found by multiplying the fraction of the U.S. population in the state times the number of House members. 8

9 That is, if P is the total U.S. population, S is the population of a given state, and H is the number of House members, then the quota for that state is S/P x H The lower quota of a state is the integer part of the quota, which is obtained by rounding the quota down to the next whole number. The upper quota is the integer you get by rounding the quota up to the next whole number. 9

10 For example, in 1790, New York s quota was Therefore, its lower quota was 9 and its upper quota is 10. For another example, in 1790 the population of Delaware was 55,540. Its quota was then 55,540 / 3,615,920 x 105 = Its lower quota was then 1 and its upper quota was 2. The following table lists the quotas for each state in

11 Apportionment of the House in 1791: Quotas State Population Quota Virginia 630, Massachusetts 475, Pennsylvania 432, North Carolina 353, New York 331, Maryland 278, Connecticut 236, South Carolina 206, New Jersey 179, New Hampshire 141, Vermont 85, Georgia 70, Kentucky 68, Rhode Island 68, Delaware 55, Totals 3,615,

12 Hamilton s Method Hamilton s method assigns to each state either their lower quota or their higher quota. The method initially assigns to each state their lower quota. We then see how many more seats need to be assigned. If there are n seats left to be assigned, then the n states with the largest fractional part of their quota receive an extra seat. 12

13 By using the lower quota for the 1791 apportionment, we have assigned 97 of the 105 seats. There are 8 seats left to assign. We then choose the 8 states with the largest fractional part of their quota. Those states are the ones listed in red in the following table. 13

14 Apportionment of the House: Fractional Parts of the quotas State Population Quota Lower Quota Fractional Part Kentucky 68, Rhode Island 68, South Carolina 206, Connecticut 236, Massachusetts 475, New York 331, Delaware 55, Pennsylvania 432, Vermont 85, Virginia 630, North Carolina 353, New Jersey 179, New Hampshire 141, Georgia 70, Maryland 278, Totals 3,615,

15 By adding one seat to each of the 8 states marked in red, we get how the house would have been apportioned in 1791 with Hamilton s method. 15

16 Apportionment of the House by Hamilton s Method State Population Quota Lower Quota Apportionment Virginia 630, Massachusetts 475, Pennsylvania 432, North Carolina 353, New York 331, Maryland 278, Connecticut 236, South Carolina 206, New Jersey 179, New Hampshire 141, Vermont 85, Georgia 70, Kentucky 68, Rhode Island 68, Delaware 55, Totals 3,615,

17 Hamilton s method wasn t used in President Washington vetoed the bill to make Hamilton s method law. His objection was that only the fractional part of the quota, and not the state s population, was used. For example, one state with a very small population may get an extra seat, while a large state may not, even though the fraction for the large state could be nearly as large as that of the small state. 17

18 Hamilton s method was adopted in 1850 and was used until However, the method has a couple problems, one of which will be addressed on Assignment 6, and the other we will discuss now. It was discovered in 1881, due to the 1880 census, which required reapportionment. Because of increases in population, the Census Bureau supplied Congress with different apportionments for a range of sizes of the house from 275 to

19 Apportionment based on 299 house seats State 1880 Population Quota Lower Quota Apportionment Alabama 1,262, Illinois 3,077, Texas 1,591, Apportionment based on 300 house seats State 1880 Population Quota Lower Quota Apportionment Alabama 1,262, Illinois 3,077, Texas 1,591,

20 Based on a 299 seat house, Alabama would have received an extra seat above their lower quota; they had the smallest fractional part of those states who received extra seats. Illinois and Texas had the largest fractional parts of the states who did not receive extra seats. Based on a 300 seat house, Alabama s fractional part, while increasing, became lower than both that of Illinois and Texas. It had the largest fractional part of any state which did not receive an extra seat. 20

21 This problem is now known as the Alabama Paradox: a state may lose a seat as the result of an increase in the house size. Increasing the house from 299 to 300 members would have resulted in Alabama going from 8 to 7 seats. 21

22 Now that the house is set at 435 members, this paradox is no longer relevant for the house. A second paradox can occur with Hamilton s method. It is the population paradox: It is possible for a state s population to increase and its apportionment decreases, while another state s population decreases and its apportionment increases. This problem will be illustrated in one of the homework problems on the next assignment. 22

23 Thomas Jefferson s Method Because President Washington vetoed the bill to make Hamilton s method of apportionment law, another method was needed. Thomas Jefferson created a method, and this method was used in the first apportionment in 1791, and was used until

24 Jefferson s method requires the choice of a divisor, a whole number d. Jefferson interpreted d as the mimimum population of a congressional district. For the 1791 apportionment, Jefferson used d = 33,000. To determine a state s apportionment with Jefferson s method, divide the state population by d, and then round down to the nearest whole number. 24

25 For example, using the 1790 census data, New York s population was 331,589. We then calculate 331,589 / d = 331,589 / 33,000 = Rounding down, New York then receives 10 seats. 25

26 Vermont s population in 1790 was 85,533. Its apportionment was then calculated as 85,533 / 33,000 = Vermont then received 2 seats, since that is what we get by rounding 2.59 down to the nearest whole number. 26

27 The divisor d in Jefferson s method needs to be selected carefully in order for the method to apportion the correct number of seats. For the 1990 population, we would need to choose d = 546,000 to come out with the correct house size of 435 members. 27

28 With d = 546,000, taking data from the 1990 census, several states would be apportioned according to the following table. Apportionment of the house with Jefferson s Method State 1990 Population Population / d Apportionment California 29,839, Montana 803, New Mexico 1,521, New York 18,044, Texas 17,059,

29 As we see below, Jefferson s method apportions the house differently than the current method used. Apportionment of the house, Jefferson s Method versus the current Hill-Huntington method State Apportionment via Jefferson s Method Actual Apportionment California Montana 1 1 New Mexico 2 3 New York Texas

30 Jefferson s method generally benefits larger states. This is possibly the reason other methods were introduced into Congress. John Quincy Adams, after stepping down as President and joining Congress as a representative of Massachusetts, introduced a method, which never was accepted. In the same year, Webster, a senator from Massachusetts, introduced his method. It took 10 years for Congress to finally approve it. 30

31 Ten years later, an Ohio representative introduced Hamilton s method. It was approved, along with a bill to change the house size to 234. This size results in both Hamilton and Webster s method giving the same apportionment. 31