Apportionment and the Census Tidbit (03/06/2000; updated 2009)

Prepared by:

Joseph Malkevitch
Mathematics and Computing Department
York College (CUNY)
Jamaica, NY 11451


web page:

The state of California has the largest number of representatives in the House of Representatives. Why? The short answer is that California has a lot more people than any other state in the United States but the long answer is much more interesting. It involves excursions into history, politics, philosophy and mathematics.

If it seems natural that California has more seats in the House of Representatives than any other state, why do California, Hawaii, Montana, and Alaska each have two Senators?

During the debate which went on in the former colonies after they managed to win a war of independence against England concerning how to govern the soon-to-be-formed new government for the United States, much attention was given to how different the former colonies were. Some of the new states would have very few people, some relatively large numbers. Some of the states were highly rural and largely agricultural, while others had big cities; commercial activities played a larger role in these states. People of wealth distrusted people with less wealth. People with more education distrusted those with less education and the distrust went in other direction, too. What was a fair way to organize the new government? Should only population play a role in the legislature for the new country or should there be some role for regional interests? Should people who were less educated or did not own property have equal voting rights as those who were more educated and did own property? Although England had a king, for nearly a hundred years power in Parliament in Britain had been growing at the expense of the King's power. Some felt that the new American government should correct some of the problems of the British Parliament which tended to under represent the interests of people flocking in increasing numbers to Britain's cities. Others wanted to put more power in the hands of people from rural areas.

The structure of the legislative branch of America's government grew out of a compromise concerning how to deal with these concerns. On the one hand each state, no matter how large, would have two Senators in one of the legislative branches of government. On the other hand, representation in the House of Representatives would be based on population.

As remarkable an achievement as the United States Constitution was, with hindsight we realize in how many ways it failed to live up to a modern view of what a democracy should try to achieve. For example, the United States Constitution accepted the notion of slavery and did not allow women the right to vote in national elections. Rather than delve into the reasons why these obviously undemocratic features became part of the Constitution, here I will discuss only aspects of the politics that have had some effect on the "mathematical issues" growing out of the Constitution's words.

Here is the wording (Article 1, Section 2 (now amended)) where it states that:

Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other persons.

Somewhat further in Section 2 it states:

The Number of Representatives shall not exceed one for every thirty thousand, but each State shall have at least one Representative;

In practical terms this means that the percentage of the total population that each state had was affected by the number of slaves who lived in the state, since slaves were not included on an equal basis with other people in counting population. Also, every state would get at least one seat, even though in some sense it seems as if this might be "unfair."

To help make some of these issues clearer we will consider some examples which do not involve 50 states and large populations but where one can see the issues involved. Suppose that a population has been arrived at for each state, and the number of seats to given out in the House of Representatives is h. Let pi denote the population of state i. Now pih is a real number which represents the exact quota that state i is entitled to out of h seats in the legislature. However, this number is almost never an integer! The actual number of seats that will be given to state i must be an integer. This issue is at the heart of the equity and mathematical problems that arise from this situation.


Suppose we have 4 states A, B, C, and D with the populations shown:

population of state A 56143

population of state B 1357

population of state C 8500

population of state D 34000

Total population: 100,000

Also, suppose there are 28 seats to distribute.

In percentage terms A has 56.14 percent of the population, B has 1.36 percent of the population, C has 8.50 percent of the population and D has 34.00 percent of the population.

The exact quotas of seats that they are entitled to are:

state A 15.72

state B .38

state C 2.38

state D 9.52

These number can be written as an integer plus a fractional part. For example, C's exact quota includes 2 whole representatives plus the fractional part .38. These numbers sum to 28. But how many seats should be given to each state? The Constitution requires that state B get at least one seat! If one were to use the usual rounding rule to generate integers in this situation, 15.72 rounds to 16, .38 rounds to zero, 2.38 rounds to 2 and 9.52 rounds to 10. If one adds these numbers up, one successfully gives away 28 seats but the trouble is that this solution is constitutionally forbidden! Now matter how small a state's exact quota, it must still get one seat according to the Constitution. This seat must come at the expense of some other state and thus be unfair! Note that if we give B 1 seat, we might decide that A gets 15, C gets 2 and D gets 9. In percentage terms: A gets 53.57 percent, B gets 3.57 percent, C gets 7.14 percent, and D gets 32.14 percent of the seats. Thus, B gets one seat as the constitution requires but all the other states as a consequence get less than their exact quota percentage share and pay for it! Note that though B and C each had the same fractional part associated with the state's exact quota, these fractional parts do not play comparable roles in the number of seats each state gets because of the requirement that each state get one a minimum of one seat.

In the United States, the number of seats that are distributed among the 50 states is 435. This number was set permanently at the time that the room in the Capitol Building in Washington was renovated to allow each of the representatives to have his/her own seat to sit in. (The British Parliament purposely has fewer physical seats available than there are people elected to the Parliament!) What happens after each state is given the number of seats that it is entitled to in the House of Representatives? Once it is decided how many seats each state gets, the legislature of that state divides up the state into districts, as many as it is entitled to seats in the House of Representatives. Often the legislature is guided by the data about where within the state people of particular political persuasion live in drawing these districts, so as to insure that the party controlling the state legislature can artificially increase the number of members of its party that get sent to Washington. This process is known as gerrymandering the districts, and although the Supreme Court has taken to curbing some of the most blatant examples of gerrymandering, the practice to some extent still goes on. Redistricting has been used to draw districts to avoid black (or other minority) representation in House of Representatives, and in recent years sometimes to guarantee members of a certain minority representation. The Supreme Court is still trying to deal with the many difficult-to--resolve equity questions that arise from the power of legislatures to draw the redistricting lines for the House of Representatives. Under usual circumstances the two major American political parties, the Democrats and the Republicans, run candidates for each of these seats. The number of Democrats and Republicans who are in the House of Representatives is the sum of the number of Democrats and Republicans who win election in the individual 435 districts. When the United States was first formed there was no explicit role for political parties in the election process. Over the years parties emerged and there have always been only two two major parties during any given period of American history. In very recent years the emergence of the Reform Party has perhaps affected politics at the national level but has not had any widespread effect in local politics in more than a few states.

Let us formulate the House of Representatives problem mathematically. Suppose that we have h objects where h is an integer and n "claimants" where claimant i is entitled to pi of h. We seek integers ai where the ai are integers that are at least one and sum to h. The intuitive idea is that ai should reflect the pi in as fair a manner as possible (even though as we have just seen one might argue that requiring that the ai be at least 1 is unfair unto itself). A mathematical "apportionment" method must be decided upon for the computation of the ai.

Before going any further it is of interest to compare how the American system of assigning states seats in the House of Representatives relates to the question of the way seats are given out in parliaments in other democracies. In most other countries when people vote for members of a parliament they do not vote for a single person in a "one member district." Rather, they vote for a party in a district where many seats (rather than one) are to be filled. In some countries, the district for which the election is being held includes all the seats. When a person votes in such an election he/she votes for a party. If the district has h seats, based on the percentage vote that each party gets the number of seats that each party gets is decided by some mathematical apportionment method. However, there is no requirement in the situation being described here that each party get one seat. It is true, however, that most countries set a threshold level of percentage vote in a district, which, if a party gets less than this threshold, it can not be considered for getting any seats. For example, it might be decided that any party getting less than 2 percent of the vote could not receive any seats. The purpose of this provision is to prevent the splinterization of the representatives in Parliament into many different parties, but it does mean that a party that nationally gets a bit short of 2 percent of the vote and might be entitled to a significant number of seats out of, say, 500 seats, would not get any.

The requirement that every state receive one seat leads to a variety of complications in the analysis to solving apportionment problems. After all, one can imagine a variety of non-electoral situations where a similar problem to apportionment arises in a natural way.


A college has been allocated 8 new positions to hire faculty. If the decision as to how many faculty will be hired by each division of the college depends on student enrollment in classes in those divisions, how many new faculty (of the 8) would each division be allocated to hire?

A company has been given permission to acquire 14 new desktop computer systems. Based on the sales of the different company divisions, how should the new systems be allocated among them?

What methods can be used to solve the apportionment problem? In our discussion below we will minimize the complications of solving apportionment problems by minimizing the issue of the complications that must be dealt with due to the requirement that every state receive at least one seat.

One early method suggested to solve the apportionment problem is based on a straightforward use of the exact quota. Give each state the integer part of its exact quota. If this does not allocate all of the seats, arrange the fractional parts in decreasing order and allocate any remaining seats in order of largest fractions. (There is the possibility of ties in such a case, but for large population numbers such ties are unlikely. The issue of what to do with ties in apportionment problems turns out to be rather critical for technical reasons that will not be specifically dealt with here.) This approach to solving apportionment problems is called Hamilton's Method or the Method of Largest Remainders. The Hamilton involved here was Alexander Hamilton, the first Secretary of the Treasury of the United States and a strong adherent for having a strong federal government in the United States.

The problem with the Method of Largest Remainders is that it has a rather unintuitive property. If one is given a collection of populations and a house size h and computes an a apportionment using largest remainder, and one now computes the apportionment with the same populations and a larger house size h', then it is possible for a state to get fewer states when h' seats are "given away" than when h seats are "given away." This situation is often referred to as the Alabama Paradox since it actually occurred in American history during a period that the House of Representatives was apportioned using the Method of Largest Remainders. The problem was resolved by continuing to raise the house size h until "the problem" disappeared. However, it was realized that Hamilton's Method was flawed in an unexpected way and as a result the method of apportioning the house of representatives was changed. (For a detailed and fascinating account of the history of the different methods to apportion the US House of Representatives, see the wonderful book of M. Balinski and H.P. Young in the references.

Early in American history a rival method for solving apportionment problems emerged, and which was championed by Thomas Jefferson. The method is best explained by an example.


Balinski, M. and H. P. Young, Fair Representation, Yale University Press, New Haven, 1982.

Malkevitch, J., et al., For All Practical Purposes (8th edition), W. H. Freeman, New York, 2009.

Back to Mathematical Modeling page