Section 15.3 Apportionment Methods

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 15.3

Apportionment Methods


INB Table of Contents

Date Topic Page #

October 16, 2013 Section 15.3 Examples 44

October 16, 2013 Section 15.3 Notes 45

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What You Will Learn Standard Divisor

Standard Quota

Lower Quota

Upper Quota

Hamilton’s Method

The Quota Rule

Jefferson’s Method

Webster’s Method

Adam’s Method

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Apportionment

The goal of apportionment is to determine a method to allocate the total number of items to be apportioned in a fair manner.

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Apportionment

Four Methods• Hamilton’s method• Jefferson’s method• Webster’s method• Adams’s method

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Standard Divisor

To obtain the standard divisor when determining apportionment, use the following formula.

total populationStandard divisor =

number of items to be allocated

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Standard Quota

To obtain the standard quota when determining apportionment, use the following formula.

population for the particular groupStandard quota =

standard divisor

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Example 1: Determining Standard QuotasThe Shanahan Law Firm needs to apportion 60 new fax machines to be distributed among the firm’s five offices. Since the offices do not all have the same number of employees, the firm’s managing partner decides to apportion the fax machines based on the number of employees at each office. Find the standard divisor given there are 1080 employees.

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Example 1: Determining Standard QuotasDetermine the standard quotas for offices B, C, D, and E of the Shanahan Law Firm and complete the table.

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Lower and Upper Quota The lower quota is the standard quota

rounded down to the nearest integer.

The upper quota is the standard quota rounded up to the nearest integer.

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Hamilton’s MethodTo 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.

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Example 2: Using Hamilton’s Method for Apportioning Fax MachinesUse Hamilton’s method to distribute the 60 fax machines for the Shanahan Law Firm discussed in Example 1.

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Office A B C D E Total

Employees 246 201 196 211 226 1080

Standard quota 13.67 11.17 10.89 11.72 12.56 60.01

Lower quota

Hamilton’s Apportionment


The Quota Rule

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

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Jefferson’s Method

1. 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.

2. Apportion to each group its modified lower quota.

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Example 4: Using Jefferson’s Method for Apportioning Legislative SeatsThe 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. Use Jefferson’s method to apportion the 250 legislature seats among the five states. The standard divisor is calculated to be 35,200.

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Webster’s Method1. 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.

2. Apportion to each group its modified rounded quota.

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Example 5: Using Webster’s Method for Apportioning Legislative SeatsConsider the Republic of Geranium and apportion the 250 seats among the five states using Webster’s method.

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Adams’s Method1. 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.

2. Apportion to each group its modified upper quota.

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

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Apportionment MethodsOf the four methods we have discussed in this section,

Hamilton’s method uses standard quotas.

Jefferson’ s method, Webster’ s method, and Adams’ s method all make use of a modified quota and can all lead to violations of the quota rule.

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Apportionment Methods

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Section 15.3 Apportionment Methods