Congressional Apportionment

What is Congressional Apportionment?

Article 1, Section 2 of the United States constitution requires the federal government to conduct a census of its population every ten years for the purpose of apportioning U.S. House of Representatives seats among the states. Each state is given a number of seats roughly equal to their population, with every state guaranteed at least one seat. Since the number of U.S. House seats is fixed at 435, a new apportionment results in some states gaining congressional seats and some states losing congressional seats.

Note that while the fairness issue is often referred to as "one-man, one-vote," the constitutional principle is strictly speaking, "one-person, one unit of representation."

How Have Congressional Seats Been Apportioned to States?

From the Census Bureau:

Since the first apportionment following the 1790 census, there have been five basic methods used to apportion the House of Representatives.

1790 to 1830 The "Jefferson method" of greatest divisors (fixed ratio with rejected fractional remainders). Under this method, a ratio of persons to representatives was selected; the population of each state was divided by that number of persons. The resulting whole number of the quotient was the number of representatives each state received. Fractional remainders were not considered, no matter how large. Thus a state with a quotient of 3.99 received three representatives, the same number as a state with a quotient of 3.01. The size of the House of Representatives was not predetermined, but resulted from the calculation.

1840 The "Webster method" of major fractions (fixed ratio with retained major fractional remainders). This method was applied in the same way as the Jefferson method, except if a fractional remainder were greater than one-half, another seat would be assigned. Thus a state with a quotient of 3.51 received four representatives, while a state with a quotient of 3.49 received three. In this method also, the size of the House of Representatives was not predetermined but resulted from the calculation.

1850-1900 The "Vinton" or "Hamilton" method established a predetermined number of representatives for each apportionment, and divided the population of each state by a ratio determined by dividing the apportionment population of the United States by the total number of representatives. The resulting whole number was assigned to each state, with an additional seat assigned, one at a time, to the states with the largest fractional remainders, up to the predetermined size of the House of Representatives. This method was subject to the "Alabama paradox," in which a state could receive fewer representatives if the size of the House of Representatives was increased.

1910, 1930 The method of major fractions assigned seats similarly to the Webster method of 1840 by rounding fractional remainders using the arithmetic mean. The ratio was selected so that the result would be the predetermined size of the House of Representatives. In 1910, the House size was fixed at 433 with provision for the addition of one seat each for Arizona and New Mexico when they became states.

1940-1990 The method of equal proportions assigns seats similarly to the Jefferson and Webster method, except it rounds fractional remainders of the quotient of the state population divided by the ratio differently. With this method, an additional seat is assigned if the fraction exceeds the difference obtained by subtracting the integer part of the quotient from the geometric mean of this integer and the next consecutive integer. The size of the House of Representatives remained fixed at 435 (except when Alaska and Hawaii became states, there was a temporary addition of one seat for each until the apportionment following the 1960 census).

Following the 1990 census, two lawsuits concerning apportionment issues were filed in federal courts. The U.S. Supreme Court held that the method of equal proportions was constitutional; that the Congress had properly exercised its apportionment authority; and that the inclusion of U.S. federal military and civilian personnel, and their dependents, in the apportionment populations of the states was constitutional. These cases were United States Department of Commerce v. Montana 112 S.Ct. 1415 (1992) and Franklin v. Massachusetts 112 S.Ct. 2767 (1992).

The Alabama Paradox

Consider a hypothetical nation composed of 14 people living in 3 states, A, B, and C such that the population is distributed as follows{A = 6, B = 6, and C = 2}. Using the Hamilton method, increasing the number of seats from 10 to 11 will result in state C losing a seat.

With 10 seats With 11 seats
State Population Fair share Seats Fair share Seats
A 6 4.286 4 4.714 5
B 6 4.286 4 4.714 5
C 2 1.429 2 1.571 1

This is called the Alabama Paradox because in 1880, the Clerk of the U.S. House discovered that increasing the size of the House of Representatives from 299 to 300 members would result in Alabama losing a seat. The actual calculation is archived here.

The Apportionment Impossibility Result

In 1982, mathematicians Michel Balinski and Peyton Young proved in their book Fair Representation: Meeting the Ideal of One Man, One Vote (this is a Google Books link to the second edition published in 2001) that any apportionment formula could not satisfy three properties:
  • The quota rule: A state receives either of the integer number of seats closest to its fair share of seats.
  • The Alabama Paradox does not exist.
  • The Population Paradox does not exists, whereby if the population size of A increases and the size of B decreases, no seats will be transferred from A to B.
See also,

John P. Mayberry. 1978.  "Quota Methods for Congressional Apportionment Are Still Non-Unique." Proc. Natl. Acad. Sci. 75(8): 3537-39.

What Are the Alternatives?

A nice analysis of the proposed alternatives is available in a 2001 Congressional Research Service Report:

David C. Huckabee. 2001. "The House of Representatives Apportionment Formula: An Analysis of Proposals for Change and Their Impact on States." Congressional Research Service Report RL31074.


Here are simulations that may be of interest:

The Current Formula: Method of Equal Proportions

An Excel spreadsheet with the current formula is available here. Follow these steps to calculate apportionment under different population scenarios:
  1. Turn on Macros
  2. Select the tab "State List"
  3. Select all (Cntl-a)
  4. Copy
  5. Paste Special - Values
  6. Point to cell A1 (upper left-hand corner)
  7. On the control panel at the top, select View-Macros (this may vary with Excel version)
  8. Select the Macro "Trans" and select run. This will create a tab "Sheet 1."
  9. Sort on Column C from largest to smallest
  10. Each of the 50 states gets 1 district. Start numbering the first row in D1 with a 51. Set D2 as a formula "D1+1" and copy and paste this formula for the D column.
  11. Scroll down to row 385 and you will see which state is on the cusp of gaining or losing a seat.
  12. Select all of the data in columns A, B, and C and Copy.
  13. Select tab "Sheet 2"
  14. Paste these data into the corresponding columns in the tab "Sheet 2"
  15. View the tab "Congressional". The counts of districts are in Column E. Note that there are comparison counts for the 2000 apportionment, and a projection based on the 2008 and 2009 census population estimates. These estimates are on the tab "NST_EST2009_ALLDATA". The data in "State List" are generated from Column O. Change the formula here to try out different population scenarios.

General Divisor Methods Simulator

A simulator for general divisor methods is here. (Requires installing the free Mathematica Player.)