1. Chapter 15: Apportionment
Part 6: Huntington-Hill Method

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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.

8. 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

9. 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 =
State
population A
453 B
367 C
697
total
1517

10. 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 =
Now we can calculate the quota for each state …
q = pi/s
State
population q A
453
453/s = B
367
367/s = C
697
697/s =
total
1517 75

11. 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 = B
367
367/s = C
697
697/s =
total
1517 75

12. 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 = B
367
367/s = C
697
697/s =
total
1517 75

13. 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 B
367
367/s = 18 C
697
697/s = 34
total
1517 75 74
in each case we round down because in each case q < q*

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

15. Example: Huntington-Hill Method
State
population q q*
ni = <<q>> A
453
453/s = 22 B
367
367/s = 18 C
697
697/s = 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 will work…

16. Example: Huntington-Hill Method
Answer
State
population q q*
<<q>> ni A
453
453/md = 22.48 22 B
367
367/md = 18.21 18 C
697
697/md = 34.59 35
total
1517 75
Here we use a modified divisor of md = 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.