Maria has taught University level psychology and mathematics courses for over 20 years. They have a Doctorate in Education from Nova Southeastern University, a Master of Arts in Human Factors Psychology from George Mason University and a Bachelor of Arts in Psychology from Flagler College.
Apportionment and Webster's Method
Fairness in representation has been a passionate issue since the origin of the Unites States of America. In fact, you might even say that the desire for fair representation caused the birth of the country. Today, states are allotted voting seats in the House of Representatives based on population. The specific calculation to determine how many seats each state should receive is called apportionment. It is easy to remember if you think about the word 'portion' in ap'PORTION'ment; it means to divide.
Over time, there have been many different proposals for how to fairly calculate each state's seat quota, the number of voting seats apportioned to the state. In this lesson, we will learn the method proposed by Daniel Webster in the 1830s. Webster's Method proposed that a divisor should be chosen such that the sum of the rounded quotas is equal to the number of seats to be apportioned.
While the proposed method may seem a bit incomprehensible, the definitions of the terms become clear in the calculations. The calculations themselves are actually pretty simple. However, the process does sometimes go around in circles. We start with standard calculations and then move on to the modified calculations, only if we need them. The sum of the quota results is the determining factor for if we need the modified versions of the calculations. Let's get to the calculations required for Webster's Method of Apportionment.
To make things easier to see, we'll start with establishing a pretend country of 1517 people and three states named State A, State B, and State C. Their populations are 453 for A, 367 for B, and C has 697 people. We want a total of 75 seats in our pretend House of Representatives. Okay, now we have all the information we need to execute Webster's Method of Apportionment.
Step 1: Divide the total population by the number of seats available. The result is called the standard divisor (SD). In our example, we would have an SD of 1517 / 75 = 20.2267.
Step 2: Divide each state population by the SD to get the standard quota (or SQ). This is the raw number of seats that should be allotted to each state. Here are the calculations of SQ for our states:
- A - SQ = 453 / 20.2267 = 22.39
- B - SQ = 367 / 20.2267 which is 18.14
- C - SQ is 697 / 20.2267 or 34.45
But, how can we have a decimal allotment? You can't really give a state four-tenths of a seat, can you? No, you can't. If you recall, Webster's Method mentioned that rounding should be used. Webster proposed traditional rounding based on the arithmetic mean. Remember that in traditional rounding, you round up if the number is 5 or greater and round down if the number is less than 5.
Looking at our standard quotas, we see that the upper quota of a standard quota is the next larger whole number. Rounding up always results in the upper quota. And the lower quota is the next smaller whole number. Rounding down always results in the lower quota. Which brings us to the next step in our process.
Step 3: Assign each state either its upper or lower quota based on traditional rounding. Back to our example, State A: SQ = 22.39, use the lower quota of 22; State B: SQ = 18.14, use a lower quota of 18; and State C: SQ = 34.45, use a lower quota of 34.
Next, we have Step 4, which is to add the quota figures together and compare the sum to the total seats to be assigned. Here, we have 22 + 18 + 34 = 74, which is not 75. We've gone through all the standard calculations, but our result is not correct. What do we do? We move on to the modified versions of these same calculations.
Okay, so our standard calculations didn't come out exact; that means we are off to the modified versions.
A modified divisor is literally a divisor chosen through trial and error in order to return exact results from the modified rounded quotas. It might seem like going in circles as you choose a divisor, run the calculations, get a result, and head back to the start to choose another divisor. Just remember to choose something close to your standard divisor.
Here's a hint: the next divisor should go in the same direction as the direction you are off the target. If the sum of the quotas is too high, then choose a larger modified divisor; if the sum is too low, choose a smaller modified divisor. The larger the number you divide by, the smaller the result and vice versa.
Choose a modified divisor for step 1 then simply repeat steps 2 - 3 until the sum of the rounded quotas equals the number of seats. Because this can go on for a while, I won't list all my modified divisors here. I'll just jump right to the one that worked and show you the calculations.
Since my sum based on the standard calculations is too low, I needed a smaller modified divisor (MD). I found that the correct MD is 20.15. With the MD chosen, we repeat the previous steps. The only difference is that we use the term 'modified' to indicate that it is not calculated with the original standard divisor.
So the modified quotas (with rounding):
- State A - MQ= 453 / 20.15 = 22.48, modified lower quota = 22
- State B - MQ = 367 / 20.15 is 18.21, modified lower quota is 18
- State C - MQ = 697 / 20.15 which equals 34.59, modified upper quota this time of 35
This shows a difference from our standard calculations. Remember, when rounding, go up if the decimal is .5 or higher and go down if it is less than .5. When we add the final rounded quotas, we get 22 + 18 + 35 = 75 and that is exactly the number of seats we needed to apportion, so we are finished.
Apportionment is simply the dividing up of something to portion out to other entities. Voting seats in the House of Representatives are apportioned through the use of calculations based on populations. Webster's Method of Apportionment is one such method proposed and adopted by the House. It states that apportioning should be accomplished through the selection of a divisor such that the ultimate traditionally-rounded quotas will sum to the exact number of seats to be assigned.
To conduct this method required the use of standard calculations. These calculations were:
- Standard divisor - total population / number of seats
- Standard quota - state population / standard divisor
- Upper quota - the closest, larger whole number to the standard quota. Rounding up results in the UQ.
- Lower quota - the closest smaller whole number to the standard quota. Rounding down results in the LQ.
If the standard calculations do not result in rounded quotas that sum to the desired number, then you must repeat the steps using a chosen modified divisor until the rounded modified quotas do sum to the exact number of seats being assigned.
Thanks for joining me. Bye!
You should have the ability to do the following after this lesson:
- Define apportionment
- Explain what the standard divisor is and what the standard quota is
- Distinguish between the upper quota and the lower quota
- Explain how to use Webster's Method of Apportionment
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