Chapter 15: Apportionment Part 6: Huntington-Hill Method

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. a 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. a b

Length is geometric mean of a and b a 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. Length is geometric mean of a and b a b

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, We define << q >> to be the result of rounding q using the geometric mean. 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. State population A 453 B 367 C 697 total 1517

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. We calculate the standard divisor first… (aka the IDEAL RATIO) In this case, s = p/h = 1517/75 = 20.2267 State population A 453 B 367 C 697 total 1517

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. We must calculate the standard divisor first. In this case, s = p/h=1517/75 = 20.2267 Now we can calculate the quota for each state … q = pi/s State population q A 453 453/s = 22.3962 B 367 367/s = 18.1444 C 697 697/s = 34.4595 total 1517 75

Example: Huntington-Hill Method Here, we are calculating q*, which is the geometric mean of the upper and lower quota for each state. State population q q* A 453 453/s = 22.3962 B 367 367/s = 18.1444 C 697 697/s = 34.4595 total 1517 75

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. State population q q* A 453 453/s = 22.3962 22.4944 B 367 367/s = 18.1444 18.4932 C 697 697/s = 34.4595 34.4963 total 1517 75

Example: Huntington-Hill Method Finally, we round based on the following rule: If the quota q is greater or equal to q* then we round up If the quota q is less than q* then we round down State population q q* ni = <<q>> A 453 453/s = 22.3962 22.4944 22 B 367 367/s = 18.1444 18.4932 18 C 697 697/s = 34.4595 34.4963 34 total 1517 75 74 in each case we round down because in each case q < q*

Example: Huntington-Hill Method State population q q* ni = <<q>> A 453 453/s = 22.3962 22.4944 22 B 367 367/s = 18.1444 18.4932 18 C 697 697/s = 34.4595 34.4963 34 total 1517 75 74 We must add a seat because the initial apportionment sums to 74 when the total house size is 75.

Example: Huntington-Hill Method State population q q* ni = <<q>> A 453 453/s = 22.3962 22.4944 22 B 367 367/s = 18.1444 18.4932 18 C 697 697/s = 34.4595 34.4963 34 total 1517 75 74 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 Answer State population q q* <<q>> ni A 453 453/md = 22.48 22.4944 22 B 367 367/md = 18.21 18.4932 18 C 697 697/md = 34.59 34.4964 35 total 1517 75 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.