01:640:103 Chap.6 Modified Divisor Methods of Apportionment
Modified Divisor Methods of Apportionment
No surplus seats to hand out
The divisor methods replace Hamilton's ranking step. Each divides every population by one shared modified divisor, rounds every modified quota by its own rule, and adjusts the divisor until the rounded quotas total exactly the number of seats.
A smaller divisor makes every quota larger, so if too few seats come out the divisor goes down, and if too many come out it goes up.
Three rounding rules
Jefferson's method rounds every modified quota down, so its divisor always sits below the standard divisor; it was used from 1791 through 1842. Webster's method rounds to the nearest whole number, and was proposed by Daniel Webster in 1832. The Huntington-Hill method, used for the House of Representatives since 1941, rounds a quota up only if it is at least the square root of its lower quota times the next whole number, so between 1 and 2 the cutoff is about 1.414 rather than 1.5.
What each method risks
Jefferson's method does not produce the Alabama paradox, but it can give a state more than its upper quota and it favours large states.
Huntington-Hill can also break the Quota Rule and leans toward small states, because its cutoff falls below the midpoint by much more for small quotas than for large ones.
In the guide's 24-bus example the largest town receives 13 buses under Jefferson although its standard quota allows only 11 or 12, and the three divisor methods produce three different apportionments.
A search procedure you can trust
Finding a working divisor is a bracketing search. Start from the standard divisor and apply the method's rounding rule.
If the seats fall short, try a smaller divisor; if there are too many, try a larger one. Keep the most recent divisor that gave too many and the most recent that gave too few, and choose the next guess between them. Report the divisor you used next to the seats, and check the total.
Because a whole interval of divisors usually works, two students can quote different divisors and both be right, provided their seats agree.
Remember too that a divisor method changes nothing about the populations or the seats; it only changes the shared divisor, so every state is treated by exactly the same rule.
What this chapter covers
- 01
Modified divisors and why they work
- 02
Jefferson's method
- 03
Webster's method
- 04
The Huntington-Hill rounding rule
- 05
Quota Rule violations and bias
Worked example · free
Webster needs a smaller divisor
- 2Standard divisor 9,000 ÷ 18 = 500; quotas 9.30, 5.36 and 3.34 round to 9, 5 and 3, only 17 seats.
- 2Too few seats, so lower the divisor. At 480 the quotas round to 10, 6 and 3, which is 19, too many.
- 2At 488 the quotas are 9.53, 5.49 and 3.42, which round to 10, 5 and 3, exactly 18.
Key terms
- Modified divisor
- A modified divisor is a divisor other than the standard one, chosen so the rounded quotas total the number of seats.
- Jefferson's method
- Jefferson's method rounds every modified quota down and uses a divisor below the standard divisor.
- Webster's method
- Webster's method rounds every modified quota to the nearest whole number, adjusting the divisor as needed.
- Huntington-Hill method
- The Huntington-Hill method rounds a quota up only when it reaches the geometric mean of its lower and upper quotas.
- Geometric mean
- The geometric mean of two numbers is the square root of their product, the Huntington-Hill cutoff.
Modified Divisor Methods of Apportionment FAQ
How do I know which way to move the divisor?
Compare the rounded total with the number of seats. Too few seats means the quotas are too small, so divide by a smaller number; too many means divide by a larger one. Each failed guess becomes a bound for the next.
Which quotas decide whether the Quota Rule is broken?
Always the standard quotas. The lower and upper quotas come from dividing by the standard divisor, even when the apportionment itself was found with a modified divisor.
Why does the Huntington-Hill method favour small states?
Its rounding cutoff is the geometric mean of the lower and upper quota, which sits noticeably below the midpoint for small quotas and barely below it for large ones. Small states therefore round up earlier than they would under Webster's method.
Exam move
Practise one data set under all three divisor methods and Hamilton, recording the divisor you used each time, then explain in one sentence why the apportionments differ.
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