In 1920, no new apportionment was done, because Congress couldn’t agree on the method to be used. They appointed a committee of mathematicians to investigate, and they recommended the Huntington-Hill Method. They continued to use Webster’s method in 1931, but after a second report recommending Huntington-Hill, it was adopted in 1941 and is the current method of apportionment used in Congress.
The Huntington-Hill Method is similar to Webster’s method, but attempts to minimize the percent differences of how many people each representative will represent.
Definition3.4.1Huntington-Hill Method
Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the standard divisor.
Divide each state’s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the quota.
Cut off the decimal part of the quota to obtain the lower quota, which we’ll call \(n\text{.}\) Compute \(\sqrt{n(n+1)}\text{,}\) which is the geometric mean of the lower quota and one value higher.
If the quota is larger than the geometric mean, round up the quota; if the quota is smaller than the geometric mean, round down the quota. Add up the resulting whole numbers to get the initial allocation.
If the total from Step 4 was less than the total number of representatives, reduce the divisor and recalculate the quota and allocation. If the total from step 4 was larger than the total number of representatives, increase the divisor and recalculate the quota and allocation. Continue doing this until the total in Step 4 is equal to the total number of representatives. The divisor we end up using is called the modified divisor or adjusted divisor.
Example3.4.2
Again, Delaware, with an initial divisor of 21,900.82927:
Figure3.4.3 Video solution by David Lippman
County
Population
Quota
Lower Quota
Geom Mean
Initial
Kent
162,310
7.4111
7
7.48
7
New Castle
538,479
24.5872
24
24.49
25
Sussex
197,145
9.0017
9
9.49
9
Total
897,934
41
This gives the required total, so we’re done.
Example3.4.4
Figure3.4.5 Video solution by David Lippman
Again, Rhode Island, with an initial divisor of 14,034.22667:
County
Population
Quota
Lower Quota
Geom Mean
Initial
Bristol
49,875
3.5538
3
3.46
4
Kent
166,158
11.8395
11
11.49
12
Newport
82,888
5.9061
5
5.48
6
Providence
626,667
44.6528
44
44.50
45
Washington
126,979
9.0478
9
9.49
9
Total
1,052,567
76
This is too many, so we need to increase the divisor. Let’s try 14,100:
County
Population
Quota
Lower Quota
Geom Mean
Initial
Bristol
49,875
3.5372
3
3.46
4
Kent
166,158
11.7843
11
11.49
12
Newport
82,888
5.8786
5
5.48
6
Providence
626,667
44.4445
44
44.50
44
Washington
126,979
9.0056
9
9.49
9
Total
1,052,567
75
This works, so we’re done.
In both these cases, the apportionment produced by the Huntington-Hill method was the same as those from Webster’s method.
Example3.4.6
Figure3.4.7 Video solution by David Lippman
Consider a small country with 5 states, two of which are much larger than the others. We need to apportion 70 representatives. We will apportion using both Webster’s method and the Huntington-Hill method.
State
Population
A
300,500
B
200,000
C
50,000
D
38,000
E
21,500
1. The total population is 610,000. Dividing this by the 70 representatives gives the divisor: 8714.286
2. Dividing each state’s population by the divisor gives the quotas
State
Population
Quota
A
300,500
34.48361
B
200,000
22.95082
C
50,000
5.737705
D
38,000
4.360656
E
21,500
2.467213
Webster’s Method
3. Using Webster’s method, we round each quota to the nearest whole number
State
Population
Quota
Initial
A
300,500
34.48361
34
B
200,000
22.95082
23
C
50,000
5.737705
6
D
38,000
4.360656
4
E
21,500
2.467213
2
4. Adding these up, they only total 69 representatives, so we adjust the divisor down. Adjusting the divisor down to 8700 gives an updated allocation totaling 70 representatives
State
Population
Quota
Initial
A
300,500
34.54023
35
B
200,000
22.98851
23
C
50,000
5.747126
6
D
38,000
4.367816
4
E
21,500
2.471264
2
Huntington-Hill Method
5. Using the Huntington-Hill method, we round down to find the lower quota, then calculate the geometric mean based on each lower quota. If the quota is less than the geometric mean, we round down; if the quota is more than the geometric mean, we round up.
State
Population
Quota
Lower Quota
Geometric Mean
Initial
A
300,500
34.48361
34
34.49638
34
B
200,000
22.95082
22
22.49444
23
C
50,000
5.737705
5
5.477226
6
D
38,000
4.360656
4
4.472136
4
E
21,500
2.467213
2
2.44949
3
These allocations add up to 70, so we’re done.
Notice that this allocation is different than that produced by Webster’s method. In this case, state E got the extra seat instead of state A.