## Chapter 15: Apportionment
**Huntington-Hill Method** ## This method is similar to both the Jefferson and Webster Methods. The Huntington-Hill, Webster and Jefferson methods are all called “divisor methods” because of the way in which a critical divisor is used to determine the apportionment. ## Like the other divisor methods, the Huntington-Hill method begins by determining a standard divisor and then calculating a quota for each state. ## Next, in the Huntington-Hill method, instead of rounding the quota in the usual way, we round to get the initial apportionments in a way that is based on a calculation involving the geometric mean of two numbers. ## Given two numbers *a* and *b*, the *geometric mean* of these numbers is
**Geometric Mean** ## We can visualize the geometric mean in two ways. Here is the first way: ## Suppose we have two numbers *a* and *b*. What is the geometric mean of these two numbers ? ## The geometric mean of *a* and *b* is the length of the side of a square who area is equal to the area of a rectangle with sides *a* and *b*.
**Geometric Mean** ## Here is another way to visualize the geometric mean of two numbers. ## Suppose we have two numbers *a* and *b*. What is the geometric mean of these two numbers ? ## The geometric mean of *a* and *b* is the length of the perpendicular segment connecting a point on the diameter of a circle with diameter length *a + b* to a point on that circle.
**Geometric Mean** ## Here is another way to visualize the geometric mean of two numbers. ## Suppose we have two numbers *a* and *b*. What is the geometric mean of these two numbers ? ## The geometric mean of *a* and *b* is the length of the perpendicular segment connecting a point on the diameter of a circle with diameter length *a + b* to a point on that circle.
**Huntington-Hill Method** ## To determine how to round q, we must calculate the geometric mean of each state’s upper and lower quota. If q is less than this geometric mean, we round down. If q is equal or greater than the geometric mean, we round up. ## Let *q** represent the geometric mean of the upper and lower quota of q. That is, ## Thus
**Huntington-Hill Method** ## Back to the Huntington-Hill Method … ## First, we calculate the standard divisor. Then we calculate q, the initial apportionment for each state. ## Next, we round q using the geometric mean method. ## Then, we determine if seats must be added or removed to result in the desired apportionment. ## If seats must be added or removed, we must choose a modified divisor, as in the Jefferson and Webster method, so that rounding the resulting quotas by the geometric-mean method will produce the required total.
**Example: Huntington-Hill Method** ## Let’s use the fictional country from a previous example. Suppose this country has states *A, B* and *C* with populations as given in the table below. Suppose the house size is 75 seats.
**Example: Huntington-Hill Method** ## Let’s use the fictional country from a previous example. Suppose this country has states *A, B* and *C* with populations as given in the table below. Suppose the house size is 75 seats.
**Example: Huntington-Hill Method** ## Let’s use the fictional country from a previous example. Suppose this country has states *A, B* and *C* with populations as given in the table below. Suppose the house size is 75 seats.
**Example: Huntington-Hill Method** ## Here, we are calculating q*, which is the geometric mean of the upper and lower quota for each state.
**Example: Huntington-Hill Method** ## Let’s use the fictional country from a previous example. Suppose this country has states *A, B* and *C* with populations as given in the table below. Suppose the house size is 75 seats.
**Example: Huntington-Hill Method** ## Finally, we round based on the following rule:
**Example: Huntington-Hill Method** ## We must add a seat because the initial apportionment sums to 74 when the total house size is 75.
**Example: Huntington-Hill Method** ## We must add a seat because the initial apportionment sums to 74 when the total house size is 75. ## With some experimentation, we find that a modified divisor of 20.15 will work…
**Example: Huntington-Hill Method** ## Here we use a modified divisor of md = 20.15. That produces modified quotas as shown above. These are compared with the geometric mean of the upper and lower modified quotas. ## Notice that with state C we round up because q is larger than q*. ## By chance, it happened that we got the same apportionment as we had using Webster’s method. Often, Webster’s method and the Huntington-Hill method will give the same result – *but not always.*
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