Apportionment

Hamilton's Method

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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] $

Hamilton's Method - Part $1$

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 = $1$

Hamilton's Method - Part $2$

Subject	Standard Quota	Decimal Part of Standard Quota	Assign
Math	$16.35$	$0.35$
English	$14.29$	$0.29$
Science	$2.36$	$0.36$	$1$

Hamilton's Method of Apportionment

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] $

Hamilton's Method - Part $1$

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 = $2$

Hamilton's Method - Part $2$

Subject	Standard Quota	Decimal Part of Standard Quota	Assign
Math	$16.8454545$	$0.8454545$	$1$
English	$14.7230303$	$0.7230303$	$1$
Science	$2.43151515$	$0.43151515$

Hamilton's Method of Apportionment

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 especially when another tutor was hired.

Alabama Paradox is the apportionment paradox that occurs when an increase in the number of items being apportioned results in a sample losing one of its initially apportioned items.
The Real Alabama Paradox accounts for this name.
Alabama, Texas, and Illinois are states in the United States of America.
In $1882$, it was found that when the House of Representatives had $299$ seats, Alabama was apportioned $8$ seats.
When the House of Representatives had $300$ seats, Alabama was apportioned $7$ seats.
The effect of adding one more seat (from $299$ to $300$) would force Alabama to give up one of its seats so that Texas and Illinois would each gain one more seat.

The Real Alabama Paradox of $1882$

	$299$ seats		$300$ seats
State	Quota	Apportionment	Quota	Apportionment
Alabama	$7.646$	$8$	$7.671$	$7$
Texas	$9.64$	$9$	$9.672$	$10$
Illinois	$18.64$	$18$	$18.702$	$19$

As if Alabama Paradox was not enough...
Let us see discuss another disadvantage of Hamilton's method

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.

Jefferson's Method

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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.

Adams's Method

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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.

Webster's Method

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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.

Lowndes's Method

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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.

Huntingdon-Hill's Method

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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:
Given a standard quota;
First: Find the lower quota
Second: Find the upper quota
Third: Calculate the geometric mean.
The geometric mean is the square root of the product of the lower quota and the upper quota.

$ GM = \sqrt{LQ * UQ} \\[3ex] $ Fourth: 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.
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 suitable divisor because we want to find a divisor that will lead to a sum of the apportioned items being equal to the number of items to apportion.
That Suitable Divisor is known as a Modified Divisor.

(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.

Dean's Method

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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:
Given a standard quota;
First: Find the lower quota

Second: Find the upper quota

Third: Find the product of the lower quota and upper quota
Multiply that product by two. This result is the numerator.

Fourth: Find the sum of the lower quota and upper quota
This result is the denominator.

Fifth: Divide the numerator by the denominator.
This result is the harmonic mean.

Sixth: 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.

(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 suitable divisor because we want to find a divisor that will lead to a sum of the apportioned items being equal to the number of items to apportion.
That Suitable Divisor is known as a Modified Divisor.

(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.

Equal Proportions Method

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Apportionment Calculator

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Hamilton's Method Calculator

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Jefferson's Method Calculator

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Adams's Method Calculator

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Webster's Method Calculator

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Lowndes's Method Calculator

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Huntingdon-Hill's Method Calculator

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Dean's Method Calculator

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Equal Proportions Method Calculator

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References

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 ‌

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