For in GOD we live, and move, and have our being.

- Acts 17:28

The Joy of a Teacher is the Success of his Students.

- Samuel Dominic Chukwuemeka

I greet you this day,

__First:__ Read the Stories (Yes, I tell stories too. ☺)

The stories will introduce you to the topic, while making you smile/laugh at the same time.

__Second:__ Review the Notes.

__Third:__ View the Videos.

__Fourth:__ Solve the questions/solved examples.

__Fifth:__ Check your solutions with my **thoroughly-explained solved examples.**

__Sixth:__ Check your answers with the calculators.

I wrote some of the codes for the calculators using Javascript, a client-side scripting language. In addition, I used the AJAX Javascript library. Please use the latest Internet browsers. The calculators should work.

Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. Thank you for visiting!!!

**Samuel Chukwuemeka** (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

**Begin with these activities/scenarios**

(1.) Note the number of students in your class.

A student may be asked to count.

Two other students can confirm the number.

(2.) **Case 1:**

The school district donated laptops to your classroom.

The laptops are to be given to your students.

The number of laptops that was donated is equal to the number of students in your classroom.

**Result:** No problem!

Each student gets a laptop.

(3.) **Case 2:**

The school district donated laptops to your classroom.

The laptops are to be given to your students.

The number of laptops that was donated is less than the number of students in your classroom.

**Result:** Hmmmm...there is a problem!

What should we do?

*
Note the responses of your students.
*

(1.) Mr. C, never mind. I have a laptop at home.

I do not need one.

But, the laptops are still fewer than the remaining number of students.

(2.) Find other ways to get more laptops.

Look for grants.

Ask wealthy philanthropists to help.

(3.)

Mr. C, I can go to the school library or public library to do my work.

I have reliable transportation.

Some do not.

You may give the laptops to them.

But, the laptops are still fewer than the remaining number of students.

(4.)

Well, as you can see; the laptops are not enough.

So, I will assign laptops to the top "number of laptops" students in the first Mathematics test.

Is this approach really fair?

It is the

What about those students who are not good in Mathematics but great in English Language?

Teacher: Alright, what do you want me to do?

(5.) Let us vote!

This is Democracy.

The top "number of laptops" students get the laptops.

However, any one who did not get a laptop may feel rejected...especially if they are children

(6.)

Remove those students who have laptops at home.

Remove the students who have reliable transportation.

For the remaining students:

Write their names in pieces of papers

Fold the papers

Place them in a container.

Ask someone to close his/her eyes and pick one at a time until the number of laptops that was given.

This is known as the List and Pick Approach of Random Sampling (Statistics)

And, we are yet to cover Introductory Statistics

So far, this seems to be the suggestion that was agreed by the Majority.

But, here is the issue.

You can do this method for a small population.

Would you do this method for a large population?

Say you have $25$ remaining students and the number of laptops is $7$:

You can use the List and Pick approach of Random Sampling

But, what if you have $300$ students?

Would you use the same approach?

Well, one can suggest that you use the Technology Approach of Random Sampling

Note other responses and write them.

Discuss the pros and cons with your students

**What does this tell us?**

There are many individuals.

There are limited resources/items.

There are many individuals competing for limited resources.

Difficult choices might be made.

Any one choice most likely will not be acceptable to everyone.

**We are learning this topic because:** we would like to know how to assign or (**apportion**) indivisible items (items that cannot be divided such as laptops) **fairly (proportionally)**
among individuals or groups.

We shall study the suggestions/methods of apportionment by some people.

We shall study the merits and demerits of these methods.

We shall discuss real-world cases where these methods are used.

Then, we shall solve applied problems using these methods.

Then, we shall try to invent any new methods that be regarded as **very fair** and **impartial**.

We are learning what other people did, so we can try to improve on their methods, or invent better methods.

NOTE: This topic only deals with __indivisible items__

For **divisible items,** there is no problem. Just divide it.

For **indivisible items,** we need to make decisions.

Welcome to **Apportionment!**

**
Bring it to the United States of America
IN GOD WE TRUST
GOD bless the United States of America.
**

**
Teacher-Student Scenario
Objective: To understand the topic of Apportionment
**

How many senators do we have?

[U.S. Constitution, Article I, section 3, clause 1]

How many House of Representatives members / Congressmen do we have?

That translates to...

Humans are indivisible

What does that imply?

How do we do the number of congressmen per state?

Or I can just review the resources you provided in the References

But, how is this assigned?

Give me a few minutes please

As of today...the 4th day of February, 2020

The State of Alabama has $7$ representatives

The State of California has $53$ representatives.

How unfair???

Before you say it is unfair, let us check their populations

As of the $1st$ day of July, $2018$

The population of the State of Alabama is $4,887,871$

While the population of the State of California is $39,557,045$

Oh...I see...

The number of representatives from each state is dependent on the population of the state.

[U.S. Constitution, Article I, section 2, clause 3]

[U.S. Constitution, Amendment XIV, section 2]

Is everyone not free?

Everyone is free.

But, the United States took several people from Africa, held them hostage, and made them to work as slaves.

By the time it was written, slavery was practiced in the United States.

Slaves were not considered free persons.

But, let us get back to the

How did they come up with those values?

Be reminded of the year it was signed and ratified...

Signed in convention September 17, 1787.

Ratified June 21, 1788.

Or what do you think should be a normal approach to assign these seats among the States?

Then, they came up with a number

Then, they asked each State to divide their population by that number.

That number is known as the

It is the

It can also be defined as the

In this case, it is the

So, we get the

Then, we divide the size of each sample (each State) by the standard divisor.

That gives us another quotient (number)

That number is known as the

Because we cannot have "decimal" people

What if we round the standard quotas to the nearest integers and we have left-overs?

What should happen to those left-overs?

In that case, it is known as

Similarly:

We can

In that case, it is known as

The

Welcome to the **Methods of Apportionment!**

**For Apportionment Problems;**

(1.) Determine the **Standard Divisor**

(2.) Determine the **Standard Quotas**

NOTE:

(a.) The **Sum of the Standard Quotas must be equal to the Number of Items to Apportion**

(b.) The **Lower Quotas, Normal Quotas, and Upper Quotas** are the rounded values of the **Standard Quotas**

(3.) Find the **Lower Quotas**

(4.) Calculate the **Sum of the Lower Quotas**

If the sum of the lower quotas is not equal to the number of items to apportion:

(5.) Find the **Normal Quotas**

(6.) Calculate the **Sum of the Normal Quotas**

If the sum of the normal quotas is not equal to the number of items to apportion:

(7.) Find the **Upper Quotas**

(8.) Calculate the **Sum of the Upper Quotas**

If the sum of the upper quotas is not equal to the number of items to apportion:

(9.) Discuss the **Methods of Apportionment**

*Please check back later for the Story * 😊

Students will:

(1.) Discuss Apportionment.

(2.) Determine the standard divisor used to apportion items.

(3.) Determine the modified divisor used to apportion items.

(4.) Determine the standard quota for each individual or sample in a population.

(5.) Determine the lower quota for each individual or sample in a population.

(6.) Determine the normal quota for each individual or sample in a population.

(7.) Determine the upper quota for each individual or sample in a population.

(8.) Apportion items among individuals or samples in a population using the Hamilton's method.

(9.) Apportion items among individuals or samples in a population using the Jefferson's method.

(10.) Apportion items among individuals or samples in a population using the Adams's method.

(11.) Apportion items among individuals or samples in a population using the Webster's method.

(12.) Apportion items among individuals or samples in a population using the Lowndes's method.

(13.) Apportion items among individuals or samples in a population using the Dean's method.

(14.) Apportion items among individuals or samples in a population using the Huntingdon-Hill's method.

(15.) Apportion items among individuals or samples in a population using the Equal Proportions method.

(16.) Solve applied problems on Apportionment.

(17.) Complete a real-world project on Apportionment.

(18.) ***Invent other fair methods of Apportionment***

**Bring it to English Language:** apportion, assign, allocate, allocation, share, distribute,
apportionment, divisible, indivisible, ratio, ration, individual, items, residue, balance

**Bring it to Mathematics:** apportionment, divisible, indivisible, dividend, divisor, quotient, remainder,
numerator, denominator, fraction, ratio, individual, sample, population, sample size,
population size, standard divisor, modified divisor, suitable divisor, standard quota,
round up, round normal, conventional rounding, round down, lower quota, normal quota, upper quota,
Hamilton's method, Jefferson's method, Webster's method, Adams's method, Huntingdon-Hill's method,
Lowndes's method, balance, residue, quota rule, apportionment paradox, Alabama paradox, population paradox,
new-states paradox, arithmetic mean, geometric mean, harmonic mean

**Bring it to Statistics:** individual, sample, population, sample size, population size

**Bring it to Computer Science:** ceiling, *ceil*, flooring, *floor*

**Apportionment** is the process of fairly assigning **indivisible items** among individuals or samples (groups).

**Population** is the entire group of individuals or thing that is being studied.

It contains all subjects of interest.

__Example:__ Halifax County in the State of North Carolina.

**Sample** is a proper subset (part) of the population being studied.

It contains some members of the population.

It contains some of the subjects of interest.

__Example:__ Halifax Community College in Halifax County.

**Individual** is a member of the population being studied.

It is a subject of interest.

__Example:__ A student in Halifax Community College.

**Standard Divisor** is **Quotient** of the **Population Size** and the **Number of Items to Apportion.**

It can also be defined as the **Average Number of People per Item**

**Standard Quota** is **Quotient** of the **Sample Size** and the **Standard Divisor.**

**Lower Quota** is the result of **rounding down** the **standard quota.**

**Normal Quota** is the result of **rounding normal (conventional rounding)** of the **standard quota.**

**Upper Quota** is the result of **rounding up** the **standard quota.**

## Symbols and Meanings

- $SD$ = Standard Divisor
- $MD$ = Modified Divisor
- $SQ$ = Standard Quota
- $LQ$ = Lower Quota
- $NQ$ = Normal Quota
- $UQ$ = Upper Quota
- $n$ = Sample Size
- $N$ = Population Size
- $NIA$ = Number of Items To Apportion
- $ANS$ = Apportioned Number per Sample
- $AM$ = Arithmetic Mean
- $GM$ = Geometric Mean
- $HM$ = Harmonic Mean

$ (1.)\:\: Standard\:\:Divisor = \dfrac{Population\:\:Size}{Number\:\:of\:\:Items\:\:to\:\:Apportion} \\[7ex] (2.)\:\: Standard\:\:Quota\:\:for\:\:Each\:\:Sample = \dfrac{Sample\:\:Size}{Standard\:\:Divisor} \\[7ex] (3.)\:\: Standard\:\:Quota\:\:for\:\:Each\:\:Sample = \dfrac{Sample\:\:Size}{Population\:\:Size} * Number\:\:of\:\:Items\:\:to\:\:Apportion \\[7ex] (4.)\:\: Lower\:\:Quota = Round\:\:down\:\:Standard\:\:Quota \\[5ex] (5.)\:\: Normal\:\:Quota = Round\:\:normal\:\:Standard\:\:Quota \\[5ex] (6.)\:\: Upper\:\:Quota = Round\:\:up\:\:Standard\:\:Quota \\[5ex] (7.)\:\: Geometric\:\:Mean = \sqrt{Lower\:\:Quota * Upper\:\:Quota} \\[5ex] (8.)\:\: Harmonic\:\:Mean = \dfrac{2 * LQ * UQ}{LQ + UQ} \\[5ex] (9.)\:\:\underline{Lowndes\:\:Method}:\:\:Number\:\:of\:\:Persons\:\:per\:\:Representative = \dfrac{Sample\:\:Size}{Lower\:\:Quota} \\[7ex] (10.)\:\:\underline{Lowndes\:\:Method}:\:\:Lowndes\:\:Ratio = \dfrac{Decimal\:\:Part\:\:of\:\:Standard\:\:Quota}{Lower\:\:Quota} $

Proposed by **Alexander Hamilton,** then Secretary of the Treasury; to apportion seats to the United States House of Representatives.

He proposed apportioning seats (items) to each State (sample) according to their lower quotas; and assigning any extra seats to the States with the largest fractions until the total number of seats is apportioned.

Also known as the **Hamilton/Vinton Method** OR the **Largest Remainder Method** OR **The Method of Largest Remainders**

Passed by Congress in $1791$ and sent to President George Washington for his signature.

President George Washington vetoed it.

Surprisingly, this was the first exercise of the Presidential veto power.

**Steps in Using Hamilton's Method:**

(1.) Calculate the standard divisor.

(2.) Calculate the standard quota for each sample.

(3.) Determine the lower quota for each sample. (Round down the standard quotas).

(4.) Add the lower quotas of all the samples.

If the sum of the lower quotas is equal to the number of items to apportion, the goal is accomplished (*not common*)

Assign the lower quotas to their respective samples.

If the sum of the lower quotas is less than the number of items to apportion (*very common*);

(a.) Assign the **lower quotas** (**integer parts** of the **standard quotas**) to their respective samples.

Extra items remain.

(b.) Assign the extra items to the samples beginning with the sample(s) with the **largest remainder** (largest decimal or fractional part) of the standard quota.

In other words, assign the extra seats to the samples in **descending order of the fractional/decimal parts** of their standard quotas.

Let us solve some examples.

**Example $1$**

Benita's Innovative School offers tutoring in Math, English, and Science.

The table shows the number of students enrolled in each subject.

The school can only afford to hire $33$ tutors .

Determine the number of tutors that should be hired for each subject using Hamilton's method.

Subject | Students |
---|---|

Math | $13,080$ |

English | $11,432$ |

Science | $1,888$ |

$ \underline{First\:\:Step} \\[3ex] Standard\:\:Divisor = \dfrac{Population\:\:Size}{Number\:\:of\:\:Items\:\:to\:\:Apportion} \\[7ex] N = 13080 + 11432 + 1888 = 26400 \\[3ex] NIA = 33 \\[3ex] SD = \dfrac{N}{NIA} \\[5ex] = \dfrac{26400}{33} \\[5ex] = 800 \\[3ex] \underline{Second\:\:Step} \\[3ex] Standard\:\:Quota\:\:for\:\:Each\:\:Sample = \dfrac{Sample\:\:Size}{Standard\:\:Divisor} \\[7ex] SQ\:\:for\:\:Math \\[3ex] = \dfrac{13080}{800} \\[5ex] = 16.35 \\[3ex] SQ\:\:for\:\:English \\[3ex] = \dfrac{11432}{800} \\[5ex] = 14.29 \\[3ex] SQ\:\:for\:\:Science \\[3ex] = \dfrac{1888}{800} \\[5ex] = 2.36 \\[3ex] $

Subject | Standard Quota | Lower Quota |
---|---|---|

Math | $16.35$ | $16$ |

English | $14.29$ | $14$ |

Science | $2.36$ | $2$ |

$\Sigma LQ = 32$ $32 \ne 33$ $33 - 32 = 1$ Balance = $1$ |

Subject | Standard Quota | Decimal Part of Standard Quota | Assign |
---|---|---|---|

Math | $16.35$ | $0.35$ | |

English | $14.29$ | $0.29$ | |

Science | $2.36$ | $0.36$ | $1$ |

Subject | Lower Quota | Extra Item | Apportion |
---|---|---|---|

Math | $16$ | $16$ | |

English | $14$ | $14$ | |

Science | $2$ | $1$ | $3$ |

$\Sigma Apportion = 33$ $NIA = 33$ $\Sigma Apportion = NIA$ |

As you can see:

Math was apportioned $16$ tutors...the lower quota

English was apportioned $14$ tutors...the lower quota

Science was apportioned $3$ tutors...the upper quota

Each subject was apportioned either the lower quota or the upper quota.

Therefore, **Quota Rule** is met.

One of the **advantages** of the **Hamilton's Method of Apportionment** is that it **satisfies** the **Quota Rule.**

So, what is the Quota Rule?

**The Quota Rule states that every sample in the population for which items are to be apportioned, should be apportioned either its lower quota or its upper quota. **

Let us discuss the reasoning behind the Quota Rule.

Do you agree that the Standard Quota signifies a fair apportionment?

*Most people agree*

Based on the fact that the Standard Quota is a true measure of a fair apportionment, it would
not seem right that any sample is apportioned more or less than one item beyond the standard quota.

In other words, any sample should be apportioned no more than one item within its standard quota.

Because the standard quota is not an integer, it would seem right that either the lower quota or the normal quota or the
upper quota is used.

The lower quota and the upper quota are the extremes/limits.

Any apportionment of items to a sample, in which the items apportioned is lower than the lower quota of that sample, violates the Quota Rule.

It is known as a **Lower Quota Violation**

Any apportionment of items to a sample, in which the items apportioned is greater than the upper quota of that sample, violates the Quota Rule.

It is known as an **Upper Quota Violation**

**Example $2$**

*Continuation from Example $1$*

Some funds were donated to Benita's Innovative School.

The school said they have the funds to hire another tutor.

Reapportion these $34$ tutors using Hamilton's method.

Subject | Students |
---|---|

Math | $13,080$ |

English | $11,432$ |

Science | $1,888$ |

$ \underline{First\:\:Step} \\[3ex] Standard\:\:Divisor = \dfrac{Population\:\:Size}{Number\:\:of\:\:Items\:\:to\:\:Apportion} \\[7ex] N = 13080 + 11432 + 1888 = 26400 \\[3ex] NIA = 34 \\[3ex] SD = \dfrac{N}{NIA} \\[5ex] = \dfrac{26400}{34} \\[5ex] = 776.470588 \\[3ex] \underline{Second\:\:Step} \\[3ex] Standard\:\:Quota\:\:for\:\:Each\:\:Sample = \dfrac{Sample\:\:Size}{Standard\:\:Divisor} \\[7ex] SQ\:\:for\:\:Math \\[3ex] = \dfrac{13080}{776.470588} \\[5ex] = 16.8454545 \\[3ex] SQ\:\:for\:\:English \\[3ex] = \dfrac{11432}{776.470588} \\[5ex] = 14.7230303 \\[3ex] SQ\:\:for\:\:Science \\[3ex] = \dfrac{1888}{776.470588} \\[5ex] = 2.43151515 \\[3ex] $

Subject | Standard Quota | Lower Quota |
---|---|---|

Math | $16.8454545$ | $16$ |

English | $14.7230303$ | $14$ |

Science | $2.43151515$ | $2$ |

$\Sigma LQ = 32$ $32 \ne 34$ $34 - 32 = 2$ Balance = $2$ |

Subject | Standard Quota | Decimal Part of Standard Quota | Assign |
---|---|---|---|

Math | $16.8454545$ | $0.8454545$ | $1$ |

English | $14.7230303$ | $0.7230303$ | $1$ |

Science | $2.43151515$ | $0.43151515$ |

Subject | Lower Quota | Extra Item | Apportion |
---|---|---|---|

Math | $16$ | $1$ | $17$ |

English | $14$ | $1$ | $15$ |

Science | $2$ | $2$ | |

$\Sigma Apportion = 34$ $NIA = 34$ $\Sigma Apportion = NIA$ |

As you can see:

Math was apportioned $17$ tutors...the upper quota

English was apportioned $15$ tutors...the upper quota

Science was apportioned $2$ tutors...the lower quota

Each subject was apportioned either the lower quota or the upper quota.

Therefore, **Quota Rule** is met.

**But, wait a minute!**

To hire $33$ tutors, Science was apportioned $3$ tutors

To hire $34$ tutors (an extra tutor), Science was apportioned $2$ tutors

So, in hiring an extra tutor; Science would have to lose a tutor just to accommodate Math and English!

**How fair is that?**

At the minimum, Science should have kept the initial apportionment of $3$ tutors.

So, this is a problem.

One of the **disadvantages** of the **Hamilton's Method of Apportionment** is that it **produces** the **Alabama Paradox.**

So, what is Alabama Paradox?

Why is it called the Alabama Paradox?

Alabama Paradox is an Apportionment Paradox

So, what is a Paradox?

So, what is Apportionment Paradox?

**A Paradox is a statement contrary to one's expectation. **

**An Apportionment Paradox is an apportionment whose results are contrary to a logical expectation**

Based on Examples $1$ and $2$; a logical expectation would be the expectation that at a minimum, $3$ tutors should be apportioned to Science.

The hiring of another tutor should not result in any loss of a Science tutor.

At the minimum, Science should have kept those $3$ tutors initially assigned to them.

**Advantages of Hamilton's Method**

(1.) Hamilton's method satisfies the Quota Rule.

It ensures that each sample is assigned either it's lower quota or upper quota.

(2.) It does not involve any trial-an-error or guesses.

(3.) Hamilton's method is fair until surplus items are added.

**Disadvantages of Hamilton's Method**

(1.) Hamilton's method is fair until surplus items are added.

Then, bigger samples are favored over smaller samples.

That is seen as preferential treatment.

(2.) Hamilton's method produces Alabama Paradox.

When at least one extra item is added, any sample should not lose any of it's initial apportionment.

Proposed by **Thomas Jefferson,** then Secretary of State; to apportion seats to the United States House of Representatives.

He proposed apportioning seats (items) to each State (sample) according to their lower quotas; and if extra seats remain, finding a suitable divisor (modified divisor) to recalculate the lower quotas.

The process will continue until all seats are apportioned. But those seats must be apportioned according to the lower quotas.

Also known as the **D'Hondt's Method** OR the **Method of Greatest Divisors**

Proposed in $1792$.

**Steps in Using Jefferson's Method:**

(1.) Calculate the standard divisor.

(2.) Calculate the standard quota for each sample.

(3.) Determine the lower quota for each sample. (Round down the standard quotas).

(4.) Add the lower quotas of all the samples.

If the sum of the lower quotas is equal to the number of items to apportion, the goal is accomplished (*not common*)

Assign the lower quotas to their respective samples.

If the sum of the lower quotas is less than the number of items to apportion (*very common*);

(a.) Assign the **lower quotas** (**integer parts** of the **standard quotas**) to their respective samples.

Extra items remain.

(b.) Use another standard divisor.

In other words, modify the standard divisor.

Or find a suitable divisor that will be used instead of the standard divisor.

We call it a **suitable divisor** because we want to find a divisor that will give lower quotas
whose sum is the number of items to apportion.

We want to find a divisor that will give lower quotas that will not give any extra item.

That **Suitable Divisor** is known as a **Modified Divisor.**

Because the lower quotas will almost always give a sum that is less than the number of items
to apportion, we have to use a smaller value of the standard divisor as the modified divisor.

This is because the division of the same number (same dividend) by smaller divisors will result in bigger quotients.

We have to keep testing several divisors until we find a modified divisor that will give us lower quotas (modified lower quotas)
whose sum is equal to the number of items to apportion.

Yes, we shall be using a trial-and-error method to find this modified divisor.

However, we shall see what we can do...as explained in the Teacher-Student Scenario (in italicized format)

Let us solve some examples.

Normally, the use of Jefferson's method will give a sum of lower quotas that is less than the
number of items to apportion.

In that case, a modified divisor is required.

A modified divisor is another value besides the standard divisor.

Because the sum of the lower quotas is less than the number of items to apportion, it is important
to select a modified divisor that is smaller than the standard divisor.

Why?

$
Say\:\: NIA = 30 \\[3ex]
Let\:\: SD = 3 \\[3ex]
Let\:\: MD = 2 \\[3ex]
\dfrac{NIA}{SD} = \dfrac{30}{3} = 10 \\[5ex]
\dfrac{NIA}{MD} = \dfrac{30}{2} = 15 \\[5ex]
10 \lt 15 \\[3ex]
$
As you can see:

Division by a smaller number gives a greater quotient.

So, we have to keep adjusting the standard divisors (changing the values of the modified divisors)
until we get a sum of lower quotas that is equal to the number of items to apportion.

*
Student: How do you know exactly what value to use...value of the modified divisor? *

Trying different values is kind of a guess work.

You said we should not be guessing in Mathematics

Teacher: Well, that is a good question.

You may want to try and discover a formula that will give you the exact modified divisor

It is worth trying.

For your tests and final exam, I shall give you a modified divisor accordingly

Or I may not ask those kind of questions.

For your homework assignments, I advise that you try some values out yourself.

And if several values do not work, you may use the calculator I developed for Jefferson's method.

You will still try some values when using the calculator.

However, there are messages that will guide you accordingly.

Besides, it is much faster.

**Advantages of Jefferson's Method**

(1.) Every sample gets at least its' lower quota.

In other words, every sample is assured of being apportioned the minimum number of items it deserves.

(2.)

**Disadvantages of Jefferson's Method**

(1.) Sometimes, Jefferson's method violates the Quota Rule.

In those cases, it produces Upper Quota violations.

In those cases, at least a sample gets an item greater than the upper quota (greater than the maximum number of items) it should have received.

(2.) It uses a trial-and-error method in determining a suitable divisor (a modified divisor) to replace the
standard divisor when the standard divisor does not lead to the correct apportionment. This is very common.

Proposed by **John Quincy Adams,** then a Representative from the State of Massachusetts; to apportion seats that will benefit New England.

He proposed apportioning seats (items) to each State (sample) according to their upper quotas; and if extra seats remain, finding a suitable divisor (modified divisor) to recalculate the upper quotas.

The process will continue until all seats are apportioned. But those seats must be apportioned according to the upper quotas.

Also known as the **Method of Smallest Divisors**

Proposed in $1832$.

**Steps in Using Adams's Method:**

(1.) Calculate the standard divisor.

(2.) Calculate the standard quota for each sample.

(3.) Determine the upper quota for each sample. (Round up the standard quotas).

(4.) Add the upper quotas of all the samples.

If the sum of the upper quotas is equal to the number of items to apportion, the goal is accomplished (*not common*)

Assign the upper quotas to their respective samples.

If the sum of the upper quotas is greater than the number of items to apportion (*very common*);

(a.) Assign the **upper quotas** (**rounding up** of the **standard quotas**) to their respective samples.

No extra items remain.

However, we cannot give what we do not have...."Nemo dat quod non habet"

There are many more seats than is available.

(b.) Use another standard divisor.

In other words, modify the standard divisor.

Or find a suitable divisor that will be used instead of the standard divisor.

We call it a **suitable divisor** because we want to find a divisor that will give upper quotas
whose sum is the number of items to apportion.

We want to find a divisor that will give upper quotas that will not give more items than is available.

That **Suitable Divisor** is known as a **Modified Divisor.**

Because the upper quotas will almost always give a sum that is greater than the number of items
to apportion, we have to use a greater value of the standard divisor as the modified divisor.

This is because the division of the same number (same dividend) by bigger divisors will result in smaller quotients.

We have to keep testing several divisors until we find a modified divisor that will give us upper quotas (modified upper quotas)
whose sum is equal to the number of items to apportion.

Yes, we shall be using a trial-and-error method to find this modified divisor.

However, we shall see what we can do...as explained in the Teacher-Student Scenario (in italicized format)

Let us solve some examples.

Normally, the use of Adams's method will give a sum of upper quotas that is greater than the
number of items to apportion.

In that case, a modified divisor is required.

A modified divisor is another value besides the standard divisor.

Because the sum of the upper quotas is greater than the number of items to apportion, it is important
to select a modified divisor that is greater than the standard divisor.

Why?

$
Say\:\: NIA = 30 \\[3ex]
Let\:\: SD = 3 \\[3ex]
Let\:\: MD = 6 \\[3ex]
\dfrac{NIA}{SD} = \dfrac{30}{3} = 10 \\[5ex]
\dfrac{NIA}{MD} = \dfrac{30}{6} = 5 \\[5ex]
10 \gt 5 \\[3ex]
$
As you can see:

Division by a bigger number gives a smaller quotient.

So, we have to keep adjusting the standard divisors (changing the values of the modified divisors)
until we get a sum of upper quotas that is equal to the number of items to apportion.

*
Student: How do you know exactly what value to use...value of the modified divisor? *

Trying different values is kind of a guess work.

You said we should not be guessing in Mathematics

Teacher: Well, that is a good question.

You may want to try and discover a formula that will give you the exact modified divisor

It is worth trying.

For your tests and final exam, I shall give you a modified divisor accordingly

Or I may not ask those kind of questions.

For your homework assignments, I advise that you try some values out yourself.

And if several values do not work, you may use the calculator I developed for Adams's method.

You will still try some values when using the calculator.

However, there are messages that will guide you accordingly.

Besides, it is much faster.

**Advantages of Adams's Method**

(1.)

(2.)

**Disadvantages of Adams's Method**

(1.) Sometimes, Adams's method violates the Quota Rule.

In those cases, it produces Lower Quota violations.

In those cases, at least a sample gets an item less than the lower quota (less than the minimum number of items) it should have received.

(2.) It uses a trial-and-error method in determining a suitable divisor (a modified divisor) to replace the
standard divisor when the standard divisor does not lead to the correct apportionment. This is very common.

Proposed by **Daniel Webster,** then a Senator from the State of Massachusetts; to apportion seats that will benefit New England.

He proposed apportioning seats (items) to each State (sample) according to their normal quotas; and if extra seats remain, finding a suitable divisor (modified divisor) to recalculate the normal quotas.

The process will continue until all seats are apportioned. But those seats must be apportioned according to the normal quotas.

It is a compromise between Jefferson's method and Adams's method.

Also known as the **Method of Major Fractions**

Proposed in $1840$.

**Steps in Using Webster's Method:**

(1.) Calculate the standard divisor.

(2.) Calculate the standard quota for each sample.

(3.) Determine the normal quota for each sample. (Round the standard quotas the conventional way/normal way).

(4.) Add the normal quotas of all the samples.

If the sum of the normal quotas is equal to the number of items to apportion, the goal is accomplished (*very common*)

Assign the normal quotas to their respective samples.

If the sum of the normal quotas is less than the number of items to apportion (*not common*);

Extra items remain.

Use another standard divisor.

In other words, modify the standard divisor.

Or find a suitable divisor that will be used instead of the standard divisor.

We call it a **suitable divisor** because we want to find a divisor that will give normal quotas
whose sum is the number of items to apportion.

We want to find a divisor that will give normal quotas that will not give any extra item.

That **Suitable Divisor** is known as a **Modified Divisor.**

We have to use a smaller value of the standard divisor as the modified divisor.

This is because the division of the same number (same dividend) by smaller divisors will result in bigger quotients.

We have to keep testing several divisors until we find a modified divisor that will give us normal quotas (modified normal quotas)
whose sum is equal to the number of items to apportion.

Yes, we shall be using a trial-and-error method to find this modified divisor.

However, we shall see what we can do...as explained in the Teacher-Student Scenario (in italicized format)

If the sum of the normal quotas is greater than the number of items to apportion (*not common*);

No extra items remain.

There are many more seats than is available.

Use another standard divisor.

In other words, modify the standard divisor.

Or find a suitable divisor that will be used instead of the standard divisor.

We call it a **suitable divisor** because we want to find a divisor that will give normal quotas
whose sum is the number of items to apportion.

We want to find a divisor that will give normal quotas that will not give more items than is available.

That **Suitable Divisor** is known as a **Modified Divisor.**

We have to use a greater value of the standard divisor as the modified divisor.

This is because the division of the same number (same dividend) by bigger divisors will result in smaller quotients.

We have to keep testing several divisors until we find a modified divisor that will give us normal quotas (modified normal quotas)
whose sum is equal to the number of items to apportion.

Yes, we shall be using a trial-and-error method to find this modified divisor.

However, we shall see what we can do...as explained in the Teacher-Student Scenario (in italicized format)

Let us solve some examples.

Normally, the use of Webster's method will give a sum of upper quotas that is greater than the
number of items to apportion.

In that case, a modified divisor is required.

A modified divisor is another value besides the standard divisor.

If the sum of the normal quotas is less than the number of items to apportion, it is important
to select a modified divisor that is smaller than the standard divisor.

Similarly, if the sum of the normal quotas is greater than the number of items to apportion, it is important
to select a modified divisor that is greater than the standard divisor.

Why?

$
Say\:\: NIA = 30 \\[3ex]
Let\:\: SD = 3 \\[3ex]
Let\:\: MD-lower = 2 \\[3ex]
Let\:\: MD-upper = 6 \\[3ex]
\dfrac{NIA}{SD} = \dfrac{30}{3} = 10 \\[5ex]
\dfrac{NIA}{MD-lower} = \dfrac{30}{2} = 15 \\[5ex]
\dfrac{NIA}{MD-upper} = \dfrac{30}{6} = 5 \\[5ex]
5 \lt 10 \\[3ex]
15 \gt 10 \\[3ex]
$
As you can see:

Division by a smaller number gives a bigger quotient...and

Division by a bigger number gives a smaller quotient.

So, we have to keep adjusting the standard divisors (changing the values of the modified divisors)
until we get a sum of normal quotas that is equal to the number of items to apportion.

*
Student: How do you know exactly what value to use...value of the modified divisor? *

Trying different values is kind of a guess work.

You said we should not be guessing in Mathematics

Teacher: Well, that is a good question.

You may want to try and discover a formula that will give you the exact modified divisor

It is worth trying.

For your tests and final exam, I shall give you a modified divisor accordingly

Or I may not ask those kind of questions.

For your homework assignments, I advise that you try some values out yourself.

And if several values do not work, you may use the calculator I developed for Webster's method.

You will still try some values when using the calculator.

However, there are messages that will guide you accordingly.

Besides, it is much faster.

**Advantages of Webster's Method**

(1.)

(2.)

**Disadvantages of Webster's Method**

(1.) Sometimes, Webster's method violates the Quota Rule.

In very rare cases, it produces either Lower Quota violations or Upper Quota violations.

(2.) It uses a trial-and-error method in determining a suitable divisor (a modified divisor) to replace the
standard divisor when the standard divisor does not lead to the correct apportionment.

Proposed by **William Lowndes,** then a Representative from South Carolina; to apportion seats to the United States House of Representatives.

He proposed apportioning seats (items) to each State (sample) according to their lower quotas; and if extra seats remain, calculating the number of persons per representative.

The extra seats will be apportioned beginning with the states with the greatest number of persons per representative.

The process will continue until all seats are apportioned.

Proposed in $1822$ as an alternative to Jefferson's method.

But, guess what?

Jefferson's method favored larger states.

His method usually favors the smaller states.

**Steps in Using Lowndes's Method:**

There are two approaches to apportionments using Lowndes's method.

**First Approach**

(1.) Calculate the standard divisor.

(2.) Calculate the standard quota for each sample.

(3.) Determine the lower quota for each sample. (Round down the standard quotas).

(4.) Add the lower quotas of all the samples.

If the sum of the lower quotas is equal to the number of items to apportion, the goal is accomplished (*not common*)

Assign the lower quotas to their respective samples.

If the sum of the lower quotas is less than the number of items to apportion (*very common*);

(a.) Assign the **lower quotas** (**integer parts** of the **standard quotas**) to their respective samples.

Some seats remain.

(b.) Calculate the number of persons per representative.

$
Number\:\:of\:\:Persons\:\:per\:\:Representative = \dfrac{Sample\:\:Size}{Lower\:\:Quota} \\[5ex]
$
The number of persons per representative is the ratio/quotient of the sample size to the lower quota.

Order the data in descending order...starting from the state with the greatest number of persons per
representative.

Assign the remaining seats in that order until all the seats are apportioned.

**Second Approach**

(1.) Calculate the standard divisor.

(2.) Calculate the standard quota for each sample.

(3.) Determine the lower quota for each sample. (Round down the standard quotas).

(4.) Add the lower quotas of all the samples.

If the sum of the lower quotas is equal to the number of items to apportion, the goal is accomplished (*not common*)

Assign the lower quotas to their respective samples.

If the sum of the lower quotas is less than the number of items to apportion (*very common*);

(a.) Assign the **lower quotas** (**integer parts** of the **standard quotas**) to their respective samples.

Some seats remain.

(b.) Calculate the Lowndes's ratio.

This is the ratio of the decimal part of the standard quota to the lower quota.

$
Lowndes\:\:Ratio = \dfrac{Decimal\:\:Part\:\:of\:\:Standard\:\:Quota}{Lower\:\:Quota} \\[5ex]
$
Order the ratios in descending order...beginning with the greatest ratio.

Assign the remaining seats in that order until all the seats are apportioned.

Use any approach you prefer.

Let us solve some examples.

Proposed by **Joseph A. Hill,** then American Statistician; and revised by **Edward V. Huntingdon,** then American Mathematician; to apportion seats to the United States House of Representatives.

Passed by the United States Congress in $1941$ and signed by President Franklin D. Roosevelt.

He proposed apportioning seats (items) to each State (sample) according to a comparison between the state's quota (standard quota) and it's geometric mean.

If the state's quota is less than it's harmonic mean, the lower quota is apportioned.

If the state's quota is greater than it's harmonic mean, the upper quota is apportioned.

Then calculate the sum to ensure that the apportioned seats is equal to the number of seats to apportion.

If the apportioned seats is not equal to the number of seats to apportion, a suitable divisor (modified divisor/adjusted standard divisor) is used as the standard divisor.

The process is repeated until all seats are apportioned.

**Steps in Using Huntingdon-Hill's Method:**

(1.) Calculate the standard divisor.

(2.) Calculate the standard quota for each sample.

(3.) Determine the lower quota for each sample. (Round down the standard quotas).

(4.) Determine the upper quota for each sample. (Round up the standard quotas).

(5.) Calculate the geometric mean for each sample.

**The Geometric Mean is the square root of the product of the lower quota and upper quota.**

(6.) Compare the standard quota for each sample with the geometric mean of that sample.

If the standard quota is less than the geometric mean, apportion the lower quota.

If the standard quota is more than the geometric mean, apportion the upper quota.

This is known as the **Huntingdon-Hill Rounding Rule**

*
Student: May you please elaborate?
Teacher: Okay. Let us review.
Given a standard quota:
Hamilton's method uses the lower quota...round down
Jefferson's method uses the lower quota...round down
Adam's method uses the upper quota...round up
Webster's method uses the normal quota/conventional quota...round the conventional way ... this is the way you have been used to
Lowndes's method uses the lower quota...round down
Huntingdon-Hill's method uses a different rounding rule...based on the comparison with the geometric mean
1st: Find the Lower quota ...round down
2nd: Find the Upper quota ...round up
3rd: Find the product of the lower quota and upper quota
Calculate the square root of that product
The result is the geometric mean.
4th: Compare the standard quota with the geometric mean.
If the standard quota is less than the geometric mean, use the lower quota.
If the standard quota is greater than the geometric mean, use the upper quota.
Student: What if the standard quota is equal to the geometric mean?
Teacher: Good question.
For any standard quota: it is not possible for that standard quota to be equal to the
geometric mean?
Student: Why?
Teacher: Give an example of a standard quota
Student: $3.574$
$
LQ = 3 \\[3ex]
UQ = 4 \\[3ex]
LQ * UQ = 3 * 4 = 12 \\[3ex]
GM = \sqrt{12} = 3.46410162 \\[3ex]
$
Student: Oh...I see
Teacher: Do you know that $\sqrt{12}$ has an indefinite value?
The $3.46410162$ is a rounded number
Student: Yes, ...a radical...an irrational number...?
Teacher: That is correct.
So, you have the standard divisor, a rational number ... being compared with it's
geometric mean, an irrational number
They can never be equal.
*

This states that:

The geometric mean is the square root of the product of the lower quota and the upper quota.

$ GM = \sqrt{LQ * UQ} \\[3ex] $

If the standard quota is less than the geometric mean, use the lower quota.

If the standard quota is greater than the geometric mean, use the upper quota.

The standard divisor can never be equal to it's geometric mean because the standard divisor is a rational number while it's geometric mean is an irrational number.

(7.) Add the apportioned items.

(a.) If the sum of the apportioned items is equal to the number of items to apportion, the goal is accomplished.

(b.) If the sum of the apportioned items is not equal to the number of items to apportion, then use another standard divisor and repeat the steps from Step $2$.

In other words, modify the standard divisor.

In other words, find a suitable divisor that will be used instead of the standard divisor.

We call it a

That

(i) If the sum of the apportioned items is smaller than the number of items to apportion, use a smaller value of the standard divisor as the modified divisor.

This is because the division of the same number (same dividend) by smaller divisors will result in bigger quotients.

(ii) If the sum of the apportioned items is greater than the number of items to apportion, use a bigger value of the standard divisor as the modified divisor.

This is because the division of the same number (same dividend) by bigger divisors will result in smaller quotients.

We have to keep testing several divisors until we find a modified divisor that will give us apportionments whose sum is equal to the number of items to apportion.

Yes, we shall be using a trial-and-error method to find this modified divisor.

However, we shall see what we can do...as explained in the Teacher-Student Scenario (in italicized format)

Let us solve some examples.

*
Student: How do you know exactly what value to use...value of the modified divisor? *

Trying different values is kind of a guess work.

You said we should not be guessing in Mathematics

Teacher: Well, that is a good question.

You may want to try and discover a formula that will give you the exact modified divisor

It is worth trying.

For your tests and final exam, I shall give you a modified divisor accordingly

Or I may not ask those kind of questions.

For your homework assignments, I advise that you try some values out yourself.

And if several values do not work, you may use the calculator I developed for Huntingdon-Hill's method.

You will still try some values when using the calculator.

However, there are messages that will guide you accordingly.

Besides, it is much faster.

Proposed by **James Dean,** then a professor of Astronomy and Mathematics at Dartmouth College; to apportion seats to the United States House of Representatives.

He proposed apportioning seats (items) to each State (sample) according to a comparison between the state's quota (standard quota) and it's harmonic mean.

If the state's quota is less than it's harmonic mean, the lower quota is apportioned.

If the state's quota is greater than it's harmonic mean, the upper quota is apportioned.

Then calculate the sum to ensure that the apportioned seats is equal to the number of seats to apportion.

If the apportioned seats is not equal to the number of seats to apportion, a suitable divisor (modified divisor/adjusted standard divisor) is used as the standard divisor.

The process is repeated until all seats are apportioned.

**Steps in Using Dean's Method:**

(1.) Calculate the standard divisor.

(2.) Calculate the standard quota for each sample.

(3.) Determine the lower quota for each sample. (Round down the standard quotas).

(4.) Determine the upper quota for each sample. (Round up the standard quotas).

(5.) Calculate the harmonic mean for each sample.

**The Harmonic Mean is the reciprocal of the arithmetic mean of the reciprocals of a data set.**

(6.) Compare the standard quota for each sample with the harmonic mean of that sample.

If the standard quota is less than the harmonic mean, apportion the lower quota.

If the standard quota is more than the harmonic mean, apportion the upper quota.

This is known as the **Dean Rounding Rule**

$
\underline{Harmonic\:\:Mean} \\[3ex]
Let\:\:LQ = L \\[3ex]
UQ = U \\[3ex]
Reciprocal\:\:of\:\:L = \dfrac{1}{L} \\[5ex]
Reciprocal\:\:of\:\:U = \dfrac{1}{U} \\[5ex]
Arithmetic\:\:Mean\:\:of\:\:the\:\:Reciprocals \\[3ex]
= \dfrac{\dfrac{1}{L} + \dfrac{1}{U}}{2} \\[7ex]
= \left(\dfrac{1}{L} + \dfrac{1}{U}\right) \div 2 \\[5ex]
\dfrac{1}{L} + \dfrac{1}{U} = \dfrac{U + L}{LU} \\[5ex]
\rightarrow \dfrac{U + L}{LU} \div 2 \\[5ex]
= \dfrac{U + L}{LU} * \dfrac{1}{2} \\[5ex]
= \dfrac{U + L}{2LU} \\[5ex]
Reciprocal\:\:of\:\:the\:\:Arithmetic\:\:Mean\:\:of\:\:the\:\:Reciprocals \\[3ex]
= \dfrac{1}{\dfrac{U + L}{2LU}} \\[7ex]
= 1 \div \dfrac{U + L}{2LU} \\[5ex]
= 1 * \dfrac{2LU}{U + L} \\[5ex]
= \dfrac{2LU}{U + L} \\[5ex]
= \dfrac{2LU}{L + U} \\[5ex]
$
*
Student: May you please elaborate?
Teacher: Okay. Let us review.
Given a standard quota:
Hamilton's method uses the lower quota...round down
Jefferson's method uses the lower quota...round down
Adam's method uses the upper quota...round up
Webster's method uses the normal quota/conventional quota...round the conventional way ... this is the way you have been used to
Lowndes's method uses the lower quota...round down
Huntingdon-Hill's method uses a different rounding rule...based on the comparison with the geometric mean...we just discussed in the previous section
Dean's method uses a different rounding rule based on the harmonic mean
1st: Find the Lower quota ...round down
2nd: Find the Upper quota ...round up
3rd: Find the product of the lower quota and upper quota
Multiply that product by two. This result is the numerator.
4th: Find the sum of the lower quota and upper quota
This result is the denominator.
5th: Divide the numerator by the denominator.
This result is the harmonic mean.
6th: Compare the standard quota with the harmonic mean.
If the standard quota is less than the harmonic mean, use the lower quota.
If the standard quota is greater than the harmonic mean, use the upper quota.
Student: What if the standard quota is equal to the harmonic mean?
Teacher: Good question.
For any standard quota: it is not possible for that standard quota to be equal to the
geometric mean?
Student: Why?
Teacher: Give an example of a standard quota
Student: $3.574$
$
LQ = 3 \\[3ex]
UQ = 4 \\[3ex]
2 * LQ * UQ = 2 * 3 * 4 = 24 \\[3ex]
LQ + UQ = 3 + 4 = 7 \\[3ex]
HM = \dfrac{24}{7} = 3.42857143 \\[3ex]
$
*
This states that:

Multiply that product by two. This result is the numerator.

This result is the denominator.

This result is the harmonic mean.

If the standard quota is less than the harmonic mean, use the lower quota.

If the standard quota is greater than the harmonic mean, use the upper quota.

(7.) Add the apportioned items.

(a.) If the sum of the apportioned items is equal to the number of items to apportion, the goal is accomplished.

(b.) If the sum of the apportioned items is not equal to the number of items to apportion, then use another standard divisor and repeat the steps from Step $2$.

In other words, modify the standard divisor.

In other words, find a suitable divisor that will be used instead of the standard divisor.

We call it a

That

(i) If the sum of the apportioned items is smaller than the number of items to apportion, use a smaller value of the standard divisor as the modified divisor.

This is because the division of the same number (same dividend) by smaller divisors will result in bigger quotients.

(ii) If the sum of the apportioned items is greater than the number of items to apportion, use a bigger value of the standard divisor as the modified divisor.

This is because the division of the same number (same dividend) by bigger divisors will result in smaller quotients.

We have to keep testing several divisors until we find a modified divisor that will give us apportionments whose sum is equal to the number of items to apportion.

Yes, we shall be using a trial-and-error method to find this modified divisor.

However, we shall see what we can do...as explained in the Teacher-Student Scenario (in italicized format)

Let us solve some examples.

__Student:__ How do you __know exactly__ what value to use...value of the modified divisor?

Trying different values is kind of a guess work.

You said we should not be guessing in Mathematics

__Teacher:__ Well, that is a good question.

You may want to try and discover a formula that will give you the exact modified divisor

It is worth trying.

For your tests and final exam, I shall give you a modified divisor accordingly

Or I may not ask those kind of questions.

For your homework assignments, I advise that you try some values out yourself.

And if several values do not work, you may use the calculator I developed for Huntingdon-Hill's method.

You will still try some values when using the calculator.

However, there are messages that will guide you accordingly.

Besides, it is much faster.

Answers are integers and/or decimals only.

Do not use a comma

__Given:__Number of Items to Apportion, Dataset (Samples and Sample Sizes)

__To Find:__Other details

Answers are integers and/or decimals only.

Do not use a comma

__Given:__Number of Items to Apportion, Dataset (Samples and Sample Sizes)

__To Find:__Other details

Answers are integers and/or decimals only.

Do not use a comma

__Given:__Number of Items to Apportion, Dataset (Samples and Sample Sizes)

__To Find:__Other details

__Given:__Number of Items to Apportion, Dataset (Samples and Sample Sizes), Modified Divisor

__To Find:__Other details

Answers are integers and/or decimals only.

Do not use a comma

__Given:__Number of Items to Apportion, Dataset (Samples and Sample Sizes)

__To Find:__Other details

__Given:__Number of Items to Apportion, Dataset (Samples and Sample Sizes), Modified Divisor

__To Find:__Other details

Answers are integers and/or decimals only.

Do not use a comma

__Given:__Number of Items to Apportion, Dataset (Samples and Sample Sizes)

__To Find:__Other details

__Given:__Number of Items to Apportion, Dataset (Samples and Sample Sizes), Modified Divisor

__To Find:__Other details

Answers are integers and/or decimals only.

Do not use a comma

__Given:__Number of Items to Apportion, Dataset (Samples and Sample Sizes)

__To Find:__Other details

Answers are integers and/or decimals only.

Do not use a comma

__Given:__Standard Quota

To round according to Huntingdon-Hill rounding rules

__Given:__Number of Items to Apportion, Dataset (Samples and Sample Sizes)

__To Find:__Other details

__Given:__Number of Items to Apportion, Dataset (Samples and Sample Sizes), Modified Divisor

__To Find:__Other details

Answers are integers and/or decimals only.

Do not use a comma

__Given:__Standard Quota

To round according to Dean rounding rules

__Given:__Number of Items to Apportion, Dataset (Samples and Sample Sizes)

__To Find:__Other details

__Given:__Number of Items to Apportion, Dataset (Samples and Sample Sizes), Modified Divisor

__To Find:__Other details

Chukwuemeka, S.D (2020, February 3). *Samuel Chukwuemeka Tutorials - Math, Science, and Technology.*
Retrieved from https://www.samdomforpeace.com

Sobecki, D., & Mercer, B. A. (2017). *Math in our World: A Quantitative Reasoning Approach.* New York, NY: Mcgraw-Hill Education.

Tannenbaum, P. (2018). *Excursions in Modern Mathematics.* Boston: Pearson.

Apportioning Representatives in the United States Congress - Dean’s Method of Apportionment | Mathematical Association of America. (2015). Maa.Org. https://www.maa.org/press/periodicals/convergence/apportioning-representatives-in-the-united-states-congress-deans-method-of-apportionment

Congressional Apportionment | US House of Representatives: History, Art & Archives. (2020). Retrieved February 4, 2020, from @USHouseHistory website: https://history.house.gov/Institution/Apportionment/Apportionment/

Gauthier, J. (2020). Apportionment Legislation 1890 - Present - History - U.S. Census Bureau. Retrieved February 4, 2020, from Census.gov website: https://www.census.gov/history/www/reference/apportionment/apportionment_legislation_1890_-_present.html

Proportional Representation | US House of Representatives: History, Art & Archives. (2019). @USHouseHistory. https://history.house.gov/Institution/Origins-Development/Proportional-Representation/

QuickFacts: Alabama. (2019). Census Bureau QuickFacts; United States Census Bureau. https://www.census.gov/quickfacts/fact/table/AL/PST045218

The 1st Article of the U.S. Constitution. (2019). National Constitution Center – The 1st Article of the U.S. Constitution. https://constitutioncenter.org/interactive-constitution/article/article-i

U.S. Senate: The Senate and the United States Constitution. (2020, January 10). Senate.Gov. https://www.cop.senate.gov/artandhistory/history/common/briefing/Constitution_Senate.htm

Young, H. (n.d.). Fairness in Apportionment. Retrieved February 7, 2020, from https://www.census.gov/history/pdf/Fairness_in_Apportionment_Young.pdf