Back To CourseMath 106: Contemporary Math
9 chapters | 106 lessons
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Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.
Have you ever wondered how the United States Congress decides how many representatives each state should have? Over the years, there have been many different methods of apportionment, that is, the method used to portion out the number of voting seats in the House of Representatives to the voting states.
In the early 1900s, Congress began using its current method of apportionment, the Huntington-Hill Method of Apportionment. In this method, geometric means are used to round modified state quotas until the sum of the modified quotas equals the exact number of seats to be apportioned in the House of Representatives.
While the definition of the method seems a bit convoluted, the calculations are quite simple. Think of it like a roller coaster. We will have to work uphill for a bit, calculating the first steps in the process. Then, have a fun trip downhill as we easily complete the next steps. After another slight uphill climb, another breezy culmination of the procedures to the finish line. However, if the final numbers do not sum to the exact number of seats being apportioned, then you must take another turn on the coaster.
Let's go through the initial steps, defining the terms as we go.
Before starting any calculations, we must first know two pieces of information: the number of seats to be apportioned and the total population. For the purposes of this review, let's assume we are apportioning 145 seats to three states with a total population of 250,000 people.
Step 1 is finding the standard divisor (SD). This is the total population divided by the number of seats to be apportioned. In our example, we have 250,000 / 145, which equals 1724.14.
Step 2 is calculating each state's standard quota (the SQ). The SQ is the state population divided by the SD. Let's identify our states as A, B, and C. State A has 125,000 people, so it would have an SQ of 125,000 / 1724.14, which is 72.499. State B has 50,000 people, giving it an SQ of 50,000 / 1724.14 = 28.99. Finally, State C has the remaining 75,000 and an SQ of 75,000 / 1724.14, which equals 43.499.
Step 3 is simply identifying the upper and lower quotas (the UQ or LQ) for each standard quota. These are the next higher and lower whole numbers, respectively. In our example, the UQs and LQs for the states are: A - 73 and 72, B - 29 and 28, and C - 44 and 43. That is an easy, downhill portion of the method procedures.
Step 4 is identifying the geometric mean for each state based on the upper and lower quotas. To find the geometric mean, you must find the square root of the product of the upper and lower quotas. For State A, the geometric mean of the quotas is the square root of 72 x 73, which equals 72.498. You can see that the geometric mean is not the same as the more traditional average, which would be half the sum of these numbers. State B has a geometric mean of 28.495 and State C has a geometric mean of 43.497.
Step 5 requires us to round each of the standard quotas based on its relative position to the geometric mean. If a state's standard quota is more than its geometric mean, then assign the upper quota. Likewise, if the standard quota is less than the geometric mean, assign the state's lower quota. In our example, State A had a standard quota of 72.499 and a geometric mean of 72.498. Thus, we round up to assign State A its upper quota of 73 seats. State B has an standard quota of 28.99 and a geometric mean of 28.495. The standard quota is greater, so, again, we round up and assign 29 seats to State B. State C has an standard quota of 43.499 and a geometric mean of 43.497, meaning we again assign the upper quota of 44.
Step 6 is the final stretch of the looped roller coaster. We need to add up all the assigned seats to find out if the sum matches the initial number of seats to be assigned. We get 73 + 29 + 44, which equals 146. And this is not equal to the number originally being assigned.
In the case that the final assigned numbers do not match the total seats to be assigned, you must go for another ride on the roller coaster. This time, instead of calculating the standard divisor, you choose a modified divisor (MD) with the intent of picking a divisor that results in the appropriate number of seats being assigned. The MD is chosen at random. However, if your total number of seats is too high, then choose a higher divisor and if your total number of assigned seats is too low, choose a lower divisor.
To save you from having to take the roller coaster ride too many times, I will give you the correct modified divisor for this example. Remember, I had to repeat steps 1 through 6 many times to find the right modified divisor. Unfortunately, that is the only way to do it. Just so you know, I had to ride the roller coaster five times before finally landing on the correct modified divisor.
So here's the secret. The correct modified divisor is 1724.25. Remember, I found this through trial and error, repeatedly following the steps above.
So, starting with State A and following the steps already discussed, I have a divisor of 1724.25, which leads to a modified quota of 125,000 / 1724.25,which equals 72.495. Notice that I am now using the term 'modified' to indicate that these calculations stem from using the randomly-picked divisor instead of the accurately calculated divisor. Continuing through the steps, we have upper and lower quotas of 73 and 72 (this hasn't changed from before), and our geometric mean is still 72.498. In this modification, we note that the MQ is less than (if only slightly) the geometric mean, so we round down and assign the lower quota of 72 seats to State A.
State B has a new modified quota of 28.9981, which is well above the geometric mean of 28.495, so we round up to the upper quota of 29. With a modified quota of 43.4971 and a geometric mean of 43.4970, State C has the closest margin, but it is still more than the mean, so we round up again, resulting in an assigned quota of 44.
Now, the sum of the assigned quotas equals exactly 145, the original number to be apportioned. Finally, the roller coaster ride is finished.
In this lesson, we reviewed the current apportionment method used in the House of Representatives. Apportionment is the method used to portion out the number of voting seats each represented state receives. The Huntington-Hill Method of Apportionment calls for a modified divisor to be found such that when using the geometric mean as a rounding basis, the sum of the assigned quotas is equal to the number of seats to be apportioned.
The steps involved calculating the following:
Each of the previous steps could be likened to a roller coaster, going uphill for the larger calculations and downhill for the easily-derived results. Just like a roller coaster, which is a continuous loop, the calculations for the Huntington-Hill Method of Apportionment continue until the final assigned quotas sum to the exact number of seats to be assigned. If the first circuit does not meet this requirement, then a modified divisor is chosen at random so that the new divisor will result in final quotas that sum to the exact number of seats to be assigned.
The amount of time you stay on the roller coaster can be a matter of luck in how quickly you land on the correct modified divisor. A tip is to move in the same direction as your final sum when you choose a new divisor. If your sum is too low, choose a lower modified divisor; if your sum is too high, choose a higher modified divisor.
Thanks for joining me on this roller coaster ride of apportionment. Bye!
When you are finished, you should be able to:
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Back To CourseMath 106: Contemporary Math
9 chapters | 106 lessons