1) The document discusses apportionment, which is the process of dividing a fixed number of representatives among groups of different sizes, such as allocating seats in the House of Representatives among US states. 2) Several apportionment methods have been used throughout US history, including Hamilton's method proposed in 1791 and Jefferson's method adopted that same year. Hamilton's method allocates seats based on population quotas and surplus seats, while Jefferson's modifies the divisor to distribute seats exactly. 3) Both methods can result in the "Alabama paradox" where a group loses a seat despite no population change, violating the principle of fairness. Later methods aimed to avoid this issue.

1. Apportionment
Prepared By: Cherie Chua

2. Apportionment
Representatives ...shall be apportioned among the several
States ...according to their respective numbers. The actual
enumeration shall be made ...every ten Years ...
- Article 1, Section 2, US Constitution
Problem: How to allocate seats among the states to the US
House of Representatives?

3. Apportionment
Apportionment is the problem of dividing up a fixed
number of things among groups of different sizes.
.
Examples:
• Dividing the seats in the U.S. House of Representatives
among the states base on the size of the population for
each state.
• nurses can be assigned to hospitals according to the
number of patients
• police officers can be assigned to precincts based on the
number of reported crimes
• math classes can be scheduled based on student demand
for those classes.

4. Apportionment
Since 1790, when the House of Representatives first
attempted to apportion itself, various methods have
been used to decide how many voters would be
represented by each member of the House.
The two competing plans in 1790 were put forward
by Alexander Hamilton and Thomas Jefferson.

5. Apportionment
Historically, at least four apportionment methods have been
implemented and the number of seats in the house has also changed
many times.
1791 The Jefferson method
Hamilton proposes his method for the first Congressional apportionment.
Washington vetoes it, and Jefferson’s method is adopted instead.
1842 The Webster method
1852 The Hamilton (Vinton) method
1901 The Webster method
Hamilton method was rejected because of the Alabama Paradox.
1941–present: The Huntington-Hill method
the House size was fixed at 435 seats and the Huntington-Hill method
became the permanent method of apportionment.

6. Apportionment
Problem: How to allocate seats among the states to the US
House of Representatives?
U.S. CONGRESS
SENATE
HOUSE OF
REPRESENTATIVES
435 SEATS 50 STATES BASED ON
EACH STATE’S
POPULATION
Apportion house representatives to US states based on state population.

7. Apportionment
The constitution mandates that every 10 years the
government produce a ‘head count’ broken down by state.
The key purpose of the state population numbers is to
meet the constitutional requirement of ‘proportional
representation.’ The Constitution requires that seats in the
House of Representatives be “apportioned among the
states according to their respective numbers” so every 10
years, two things happen:
1. The state population must be determined (by Census) and
2. The seats in the House of Representatives must be
apportioned to the states based on their populations.

8. Apportionment – Hamilton Method
Hamilton’s Method
In 1791 Alexander Hamilton proposed the following simple
method as a way to apportion the US House of Representatives.
1. Compute the standard divisor
2. Calculate and round down each state’s standard quotas
3. Distribute the surplus seats, one per state, starting with
the largest leftover fractional part, then proceeding to
the next largest, and so on, until all the surplus seats
have been dealt with.

9. Apportionment – Hamilton Method
Hamilton’s Method
Hamilton’s method can be described quite briefly:
Every state gets at least its lower quota. As many states
as possible get their upper quota, with the one with
highest residue (i.e.,fractional part) having first priority,
the one with second highest residue second priority, and
so on.

10. Apportionment – Hamilton Method
standard divisor =
total population
number of people to apportion
standard quota =
population
standard divisor
The standard quota is the whole number part of the
quotient of a population divided by the standard divisor.

11. Apportionment – Hamilton Method
Example 1 The Congress of Power
Parador is a small republic located in Central America
and consists of five states. There are 25 seats in the
Congress which are to be apportioned among the
five states to their respective population.
State Population
Apus 11,123
Libra 879
Draco 3,518
Cephus 1,563
Orion 2,917
20,000

12. Apportionment – Hamilton Method
20,000
25
standard quota =
population
standard divisor
Standard divisor = 800
State Population
Apus 11,123
Libra 879
Draco 3,518
Cephus 1,563
Orion 2,917
11123
800
= 13.90
879
800
= 1.10
3518
800
= 4.40
1563
800
= 1.95
2917
800
= 3.65
13
1
4
1
3
25 seats
22
14
1
4
2
4
Quotient Standard
Quota
Number of
Representatives
Lower Quota

13. Apportionment – Jefferson Method
Jefferson Method
In the previous example, we were three representatives
short.
The apportionment method suggested by future president
Thomas Jefferson as a competitor to Hamilton's method.
Jefferson's method was the first apportionment method
used by the US Congress starting at 1791 through 1842 when
it was replaced by Webster's method.

14. Apportionment – Jefferson Method
Jefferson Method
1. Compute the standard divisor
2. Decrease the standard divisor by an amount such
that when state allocations are rounded downward,
they add up to the exact number of seats.
Jefferson’s method divides all populations by a modified
divisor and then rounds the results down to the lower
quota. We keep guessing modified divisors until the method
assigns the correct total number of seats. Our guess for the
first modified divisor should be a number smaller than the
standard divisor.

15. Apportionment – Jefferson Method
20,000
25
Standard divisor = 800
State Population Quotient Number of
Representatives
Apus 11,123
Libra 879
Draco 3,518
Cephus 1,563
Orion 2,917
11123
750
= 14.83
879
750
= 1.17
3518
750
= 4.69
1563
750
= 2.08
2917
750
= 3.89
24 seats
14
1
4
2
3
Modified standard divisor = 750

16. Apportionment – Jefferson Method
20,000
25
Standard divisor = 800
State Population Quotient Number of
Representatives
Apus 11,123
Libra 879
Draco 3,518
Cephus 1,563
Orion 2,917
11123
740
= 15.03
879
740
= 1.19
3518
740
= 4.75
1563
740
= 2.11
2917
740
= 3.94
25 seats
15
1
4
2
3
Modified standard divisor = 740

17. Fairness in Apportionment
One criterion of fairness for an apportionment plan is that it
should satisfy the quota rule.
Quota Rule
The number of representatives apportioned to a state
is the standard quota or one more than the standard
quota

18. Fairness in Apportionment
The Jefferson method, does not always satisfy the quota rule
by calculating the standard quota of Apus
The standard quota of Apus is 13. However, the
Jefferson method assigns 15 representatives to that
state, two more than its standard quota. Therefore, the
Jefferson method violates the quota rule.

19. Apportionment
Example 2
Suppose the 18 senior police officers in a city that
are employed are selected to patrol the five districts
according to the populations of the five districts.
DIstrict Population
A 7,020
B 2,430
C 1,540
D 3,720
E 5,290
Determine the number of
police officers in each city
should have by:
a. Hamilton method
b. Jefferson method
20,000

20. Apportionment – Hamilton Method
20,000
18
standard quota =
population
standard divisor
Standard divisor = 1111.11
District Population Quotient Standard
Quota
Number of Senior
Police Officers
A 7,020
B 2,430
C 1,540
D 3,720
E 5,290
7020
1111.11
= 6.32
2430
1111.11
= 2.19
1540
1111.11
= 1.39
3720
1111.11
= 3.35
5290
1111.11
= 4.76
6
2
1
3
4
18
16
6
2
2
3
5

21. Apportionment – Jefferson Method
20,000
18
Standard divisor = 1111.11
District Population Quotient Number of Senior
Police Officers
A 7,020
B 2,430
C 1,540
D 3,720
E 5,290
7020
925
= 7.59
2430
925
= 2.63
1540
925
= 1.67
3720
925
= 4.02
5290
925
= 5.72
19
7
2
1
4
5
Modified standard divisor = 925

22. Apportionment – Jefferson Method
20,000
18
Standard divisor = 1111.11
District Population Quotient Number of Senior
Police Officers
A 7,020
B 2,430
C 1,540
D 3,720
E 5,290
7020
950
= 7.39
2430
950
= 2.56
1540
950
= 1.62
3720
950
= 3.92
5290
950
= 5.57
18
7
2
1
3
5
Modified standard divisor = 950

23. Apportionment – Hamilton Method
Example 3
Suppose the 18 senior police officers in a city that
are employed are selected to patrol the five districts
according to the populations of the five districts.
District Population
A 7,020
B 2,430
C 1,540
D 3,720
E 5,290
Suppose that the department
head decides to add one more
senior police officer even though
the population of each city
remains the same. Determine how
the members of the board will be
apportioned.
20,000

24. Apportionment – Hamilton Method
20,000
19
standard quota =
population
standard divisor
Standard divisor = 1052.63
District Population Quotient Standard
Quota
Number of Senior
Police Officers
A 7,020
B 2,430
C 1,540
D 3,720
E 5,290
7020
1052.63
= 6.67
2430
1052.63
= 2.31
1540
1052.63
= 1.46
3720
1052.63
= 3.53
5290
1052.63
= 5.03
6
2
1
3
5
19
17
7
2
1
4
5

25. Apportionment – Hamilton Method
District Hamilton Apportionment with
18 Senior Police Officers
Hamilton Apportionment with
19 Senior Police Officers
A 6 7
B 2 2
C 2 1
D 3 4
E 5 5
Note that although one more police officer was added, district C
lost a police officer, even though the populations of the district
did not change. This is called the Alabama paradox. It has a
negative effect on fairness.

26. Apportionment – Hamilton Method
The flaw of Hamilton Method
The Alabama paradox occurs when an increase in the total
number of items to be apportioned results in a loss of an item
for a group.
At first glance, Hamilton’s method appears to be quite fair.
It could be reasonably argued that Hamilton’s method has a
major flaw in the way it relies entirely on the size of the
residues without consideration of what those residues
represent as a percent of the state’s population.

27. Apportionment – Hamilton Method
Example 4
The number of students taking mathematics, English, and science
courses during the 1st and 2nd semesters. If 100 full-time teaching
positions are to be apportioned among the three departments on
the basis of their respective course enrollments.
1st semester
Subject Population
Math 951
English 1949
Science 7100
10,000 10,030
2nd semester
Subject Population
Math 962
English 1968
Science 7100
is the apportionment fair? Explain your reasoning

28. Apportionment – Hamilton Method
1st semester
Subject Population Quotient Standard
Quota
Number of
teachers
Math 951
English 1949
Science 7100
10,000
Standard
divisor
10,000
100
= 100
951
100
= 9.51
1949
100
= 19.49
7100
100
= 71
9
19
71
99
10
19
71
100
2nd semester
Subject Population Quotient Standard
Quota
Number of
teachers
Math 962
English 1968
Science 7100
10,030
Standard
divisor
10,030
100
= 100.3
962
100.3
= 9.59
1968
100.3
= 19.62
7100
100.30
= 70.79
9
19
70
98
9
20
71
100

29. Apportionment – Hamilton Method
The rate of growth for Math
962-951
951
= 1.16%
=
11
951
The rate of growth for English
1968-1949
1949
= 0.97%
=
19
1949
Thus, mathematics was
growing at a faster rate
(1.16%) than English
(0.97%), but despite this,
mathematics lost 1 position
(from 10 to 9), whereas
English gained 1 position
(from 19 to 20).
The population paradox occurs when the population of
group A is increasing faster than the population of
group B, yet A loses items to group B.

30. Apportionment – Huntington-Hill Method
Huntington-Hill Method
This method has been used since 1941.
The Huntington-Hill Method is the current method
Congress uses to apportion the U.S. House of
Representatives. This method was originally proposed
in 1911 by Joseph Hill, the Chief Statistician of the
Bureau of the Census, and later improved and refined
by Edward Huntington, a Professor of Mechanics and
Mathematics at Harvard University.

31. Apportionment – Huntington-Hill Method
Huntington-Hill Method
1. Find the standard divisor
2. Calculate each state’s quotas
3. Calculate the geometric mean of each state
4. If the quota is larger than the geometric mean, round up; if the
quota is smaller than the geometric mean, round down. Add up
the resulting whole numbers to get the initial allocation.
5. If the total was less than the total number of representatives,
reduce the divisor and recalculate the quota and allocation. If the
total was larger than the total number of representatives,
increase the divisor and recalculate the quota and allocation.
Continue doing this until the total is equal to the total number of
representatives.
G = a×b

32. Apportionment – Huntington-Hill Method
Example 1
Suppose the 18 senior police officers in a city that
are employed are selected to patrol the five districts
according to the populations of the five districts.
DIstrict Population
A 7,020
B 2,430
C 1,540
D 3,720
E 5,290
Determine the number of
board members in each city
should have by Huntington-Hill
Method
20,000

33. Apportionment – Huntington-Hill Method
20,000
18
standard quota =
population
standard divisor
Standard divisor = 1111.11
District Population Quotient Lower
Quota
Geometric
Mean
Number of Senior
Police Officers
A 7,020
B 2,430
C 1,540
D 3,720
E 5,290
7020
1111.11
= 6.318
2430
1111.11
= 2.187
1540
1111.11
= 1.386
3720
1111.11
= 3.348
5290
1111.11
= 4.761
6
2
1
3
4
6.481
2.449
1.414
3.464
4.472
6
2
1
3
5
17

34. Apportionment – Huntington-Hill Method
Modified divisor = 1085
District Population Quotient Lower
Quota
Geometric
Mean
Number of Senior
Police Officers
A 7,020
B 2,430
C 1,540
D 3,720
E 5,290
7020
1085
= 6.47
2430
1085
= 2.240
1540
1085
= 1.419
3720
1085
= 3.429
5290
1085
= 4.876
6
2
1
3
4
6.481
2.449
1.414
3.464
4.472
6
2
2
3
5
18

35. Apportionment – Huntington-Hill Method
Huntington-Hill Method
When the Huntington-Hill method is used to
apportion representatives between two states, the
state with the greater Huntington-Hill number
receives the next representative. This method can
be extended to more than two states.
Huntington-Hill Apportionment Principle
When there is a choice of adding one representative to
one of several states, the representative should be added
to the state with the greatest Huntington-Hill number.

36. Apportionment – Huntington-Hill Method
Huntington-Hill Method
The value of , where PA is the population of the
state A and a is the current number of representatives
from state A, is called the Huntington-Hill number for
state A.
PA
( )
2
a a+1
( )

37. Apportionment – Huntington-Hill Method
The table below shows the numbers of lifeguards that are
assigned to three different beaches and the numbers of
rescues made by lifeguards at those beaches. Use the
Huntington-Hill apportionment principle to determine to
which beach a new lifeguard should be assigned.
Beach Number of rescues Number of Lifeguards
A 1227 37
B 1473 51
C 889 24
Example 2

38. Apportionment – Huntington-Hill Method
Beach Number of rescues Number of Lifeguards
A 1227 37
B 1473 51
C 889 24
Example 3
Calculate the Huntington-Hill number for each of the beaches.
PA
( )
2
a a+1
( )
Beach A
1227
( )
2
37 37+1
( )
= 1071
Beach B
1473
( )
2
51 51+1
( )
= 818
Beach C
889
( )
2
24 24+1
( )
= 1317
The new lifeguard should be assigned to Beach C.

39. Basic Concept and Terminology
States - is the term we will use to describe the
parties having a stake in the apportionment.
Seats - This term describes the set of M identical,
indivisible objects that are being divided among the
N states. For convenience, we will assume that
there are more seats than there are states, thus
ensuring that every state can potentially get a seat.

40. Basic Concept and Terminology
Population - his is a set of N positive numbers (for
simplicity we will assume that they are whole
numbers) that are used as the basis for the
apportionment of the seats to the states.
The standard divisor(SD)
This is the ratio of population to seats.
The standard quota of a state is the exact
fractional number of seats that the state would get
if fractional seats were allowed.

41. Basic Concept and Terminology
Upper and lower quotas
The lower quota is the standard quota rounded
down and the upper quota is the standard quota
rounded up.
apportionment method–a reliable procedure that
(1) will always produce a valid apportionment
(exactly M seats are apportioned) and (2) will
always produce a “fair” apportionment.