United States congressional apportionment
The membership of the United States House of Representatives changes each decade following the decennial United States Census. Each state is apportioned a number of members of Congress based upon its population. This number also determines the state's number of electors in presidential elections, which equals the size of their congressional delegation (House plus Senate).
House size
The United States Constitution requires that
 The number of Representatives shall not exceed one for every thirty thousand, but each state shall have at least one Representative (Article I, Section 2, Paragraph 3)
Prior to the twentieth century, the number of representatives increased every decade as more states joined the union, and the population increased.
In 1911, Public Law 62-5 set the membership of the U.S. House at 433; with the subsequent admission of Arizona and New Mexico as states, membership increased to 435, where it has remained (except for a brief period from 1959 to 1963 following the admission of Alaska and Hawaii, during which House membership was 437). If the ratio as specified by the Constitution of one representative for every 30,000 people were maintained today, the House of Representatives would have about 9,500 members.
Apportionment methods
Apart from the fact that the number of delegates is at least one for each state, as required by the Constitution, this number is in principle proportional to population (equalizing the size of congressional districts nationwide). To arrive at whole numbers, the Method of Equal Proportions is used. The method first assigns one seat to each state, and then assigns each additional seat successively to the state with the highest "priority value", a value for the population per seat. For the latter the question would arise whether the current number of seats or one more should be taken. This is solved by taking an intermediate value, the geometric mean of the two. The resulting priority value is the geometric mean of the current population per seat and the population per seat in the case the state gets the extra seat.
Computing for every state and any number of seats the priority value, and sorting the list in descending order of the resulting values, the first 385 are applicable (seats 51-435) (see Census 2000 Ranking of Priority Values).
The Equal Proportions method has been the fifth distinct method of determining congressional apportionment since the adoption of the United States Constitution. The size of the Congressional delegations from the thirteen original states were assigned by the Constitution for use until the completion of the first U.S. Census. Legislation admitting new states into the union has also designated the number of representatives of states until the time of the next census.
The Equal Proportions Method
Reapportionment of the United States House of Representatives (the lower house of the U.S. Congress) occurs every year ending in "1", the year after the U.S. Census Bureau performs the decennial census mandated by the U.S. Constitution. Once seats have been reapportioned to the various states, each state creates districts of approximately equal population, a process called redistricting.
The United States authorizes there to be 435 members of Congress to be divided up between the 50 states. The seats are "apportioned" out to each state. Apportionment is done each time a new census is taken and they figure out how many people live in a particular state. So the total number of representatives is 435, and they are apportioned out to each state based on the state's population.
Apportionment is done through the Equal Proportions Method.
First, each state is automatically guaranteed at least one seat in Congress. That means there are a total of 385 seats left to hand out.
The remaining seats are handed out one at a time, to the state that "deserves" another seat the most. Thus, the 51st seat always goes to the largest state (currently California).
Now who gets the 52nd? We need to somehow take into acount the fact that California already has a second seat, and so "deserves" a third one less. That is, we need a mathematical formula that expresses the priority formula that a state should have for getting another seat.
The formula used by the method of equal proportions is
where P is the population of the state, and n is the number of seats it currently has.
When all states have 1 seat, the largest value of A corresponds to the largest state. But now that California has 2 seats, its priority value decreases, and it has to take a step back in line. The 52nd seat goes to Texas, the 2nd largest state, but the 53rd goes back to California, and so on until all the seats have been handed out. Each time a state gets a seat, its priority drops and another state comes to the top of the list.
The Census 2000 Ranking of Priority Values (see link above) shows the order in which seats 51-435 were apportioned after the 2000 Census, with additional listings for the next five priorities. North Carolina was allocated the final (435th) seat. Utah (priority list 436) missed a fourth seat by only 857 residents. Legal action by Utah to amend the results, citing irregularities in North Carolina and undercounting of Utah's overseas population, was unsuccessful.
Notes
Delegate counts in italics represent temporary counts assigned by Congress until the next decennial census or by the U.S. Constitution in 1789 until the first U.S. Census.
Elections held in the year of a census use the apportionment determined by the previous census.
* The state of Maine was formed out of portions of Massachusetts in 1820.
** The state of West Virginia was formed out of portions of Virginia in 1863.
See also
List of states ordered by number of electors in the presidential elections, which is two more for each state.
United States Congress
United States House of Representatives
Alabama paradox
Apportionment paradox
Redistricting
Partisan mix of congressional delegations
External links
Congressional Apportionment by the U.S. Census Bureau
Cut-the-knot.org:
* The Constitution and Paradoxes
* A Java Simulation of Adams' method
* A Java Simulation of Hamilton's method
* A Java Simulation of the Huntington-Hill method
* A Java Simulation of Jefferson's method
* A Java Simulation of Webster's method
Thirty-thousand.org:
* A Brief History of Apportionment
* Forty or Thirty Thousand?
* The Size of the U. S. House of Representatives and its Constituent State Delegations Authorized Number of Seats by Year and by Congress 1789 to 2006
* Outcomes of Presidential Elections and the House Size
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