Method of Equal Proportions: Priority Values and the Geometric Mean

The Method of Equal Proportions is the formula the federal government uses to divide the 435 voting seats in the U.S. House of Representatives among the 50 states after each decennial census. It starts by giving every state one seat, then awards the remaining 385 one at a time, each seat going to whichever state has the highest “priority value” at that moment. The priority value is the state’s population divided by the geometric mean of its current seat count and the next one up. The method is required by federal statute, 2 U.S.C. §2a, which directs the President to transmit apportionment figures to Congress using “the method known as the method of equal proportions.”¹

The Starting Point: One Seat Per State

Before any calculation runs, each of the 50 states receives one seat. That guarantee is written into the same statute that mandates the formula, and it accounts for the first 50 of the 435 seats.¹ The math only decides where the remaining 385 seats go.

The overall size of the House was fixed at 435 by the Apportionment Act of 1911 and has stayed there ever since, apart from a temporary bump to 437 when Alaska and Hawaii joined the union.² Because the total is capped, apportionment is a zero-sum exercise: any state that gains a seat does so because another state lost one.

The Priority Value Formula

For every state, at every seat level it could plausibly reach, the Census Bureau calculates a priority value using this formula:

$P = V / \sqrt{n(n+1)}$

$V$ is the state’s apportionment population. $n$ is the number of seats the state currently holds. Since every state already has one seat before the ranking begins, the first calculation for each state uses $n = 1$, producing the priority value for its potential second seat.³

The denominator is the geometric mean of $n$ and $n+1$. As a state accumulates more seats, that denominator grows, and its priority value for the next seat falls. A state’s claim to a second seat (denominator $\sqrt{2} \approx 1.414$) is far stronger than its claim to a twentieth (denominator $\sqrt{380} \approx 19.49$). The formula bakes in diminishing returns: each additional seat is harder to earn than the last.

How the 385 Remaining Seats Are Assigned

The Bureau builds a single ranked list of every priority value from every state. Seat 51, the first seat beyond the guaranteed 50, goes to whichever state sits at the top of that list.³ That state’s next priority value, now calculated with $n = 2$ for a possible third seat, takes its place in the ranking. Seat 52 goes to the new leader.

The process repeats until all 435 seats are assigned. The most populous states appear near the top of the list repeatedly and pick up several seats in a row before smaller states climb into contention. But each round recalculates, so no state runs the table indefinitely. Once seat 435 is assigned, any priority value that ranked 436th or lower is simply left on the table for that decade.

Why the Geometric Mean

Using the geometric mean rather than the arithmetic mean is what distinguishes this method from its predecessors. The geometric mean of two consecutive numbers is always slightly lower than their arithmetic average. The geometric mean of 4 and 5, for example, is about 4.47, not 4.5.

That difference sets a slightly higher bar for rounding a state up to another seat, and it produces apportionments that minimize the percentage difference in district populations between any two states. Methods that use the arithmetic mean or the raw quotient tend to minimize absolute differences instead, which quietly favors larger states. Focusing on relative difference gives smaller states a marginally fairer shot, though no method eliminates the gap between the largest and smallest districts entirely.

Congress didn’t arrive at this choice quickly. Through the 1800s and early 1900s the House cycled through Hamilton’s, Jefferson’s, and Webster’s methods, each producing outcomes that felt unfair or generated paradoxes such as the Alabama Paradox, in which enlarging the House could cause a state to lose a seat. In 1929 the Speaker asked the National Academy of Sciences to study the problem. A committee of four mathematicians recommended the method developed by Edward V. Huntington, and a later NAS panel that included John von Neumann confirmed it. Congress formally adopted the method in 1941.⁴

What Goes Into the Population Number

The $V$ in the formula is the apportionment population, and it is broader than a voter roll. It includes every person residing in the United States on Census Day, citizens and noncitizens alike, counted at their usual place of residence.⁵ The constitutional language calls for an enumeration of persons, not citizens.

One group is added on top of resident counts. U.S. military personnel and federal civilian employees stationed overseas, along with their dependents, are allocated to their home state using the employer’s administrative records.⁶ These overseas counts are used only for apportionment, not for drawing district lines, and for a state with a heavy military presence the adjustment can shift the seat total.

How Close the Margins Get: The 2020 Cycle

The final seats on the list often come down to very small numbers. In the 2020 apportionment, Minnesota received the 435th seat. New York held the 436th priority value and lost a seat as a result. The Census Bureau calculated that New York would have kept that seat with just 89 more people in its count, assuming every other state’s population stayed the same.⁷

The broader 2020 results showed the formula in action. Texas gained two seats. Colorado, Florida, Montana, North Carolina, and Oregon each gained one. California, Illinois, Michigan, New York, Ohio, Pennsylvania, and West Virginia each lost one.⁷ California’s loss was the first in its history as a state. Montana, represented by a single at-large member since 1993, returned to two districts.

If Congress Does Nothing

The system is designed to run without new legislation. If Congress fails to pass an apportionment law after a census, the method of equal proportions still governs, and each state keeps the seat count from the most recent presidential apportionment statement.¹ That automatic mechanism, added by the Permanent Apportionment Act of 1929, exists because after the 1920 census political gridlock blocked any reapportionment at all and the House kept using the 1910 numbers for a decade.

What Apportionment Does Not Do

The formula only decides how many seats each state gets. It does not draw district lines. Redistricting is a separate, state-level process that starts once the Census Bureau delivers detailed population data to the states, and each state handles it through its own legislature, an independent commission, or a hybrid arrangement.⁸

The method also applies only to the 50 states. U.S. territories and the District of Columbia are not part of the apportionment population and do not receive voting House seats; their residents are represented by non-voting delegates or, for Puerto Rico, a Resident Commissioner.⁹

1
Office of the Law Revision Counsel. 2 USC 2a – Reapportionment of Representatives
2
United States Census Bureau. Historical Perspective
3
United States Census Bureau. Methods of Apportionment
4
Office of the Law Revision Counsel. 2 USC 2b – Number of Representatives From Each State in 78th and Subsequent Congresses
5
United States Census Bureau. Frequently Asked Questions
6
United States Census Bureau. 2020 Census Apportionment Results – Table A
7
United States Census Bureau. 2020 Census Apportionment Results – Table D
8
United States Census Bureau. Decennial Census P.L. 94-171 Redistricting Data
9
Office of the Law Revision Counsel. 48 USC Chapter 16 – Delegates to Congress