One Person, One Partial Vote

Achieving One-Person-One-Vote equality in the federal House.

As explained in Section Seven of this web pamphlet, the population sizes of congressional districts vary significantly from state to state as a result of Congress allowing far too few Representatives relative to the nation’s total population. Moreover, those Americans living in the larger (more populous) districts are politically disadvantaged compared to those living in the smaller (less populous) districts. Consequently, the House of Representatives is in egregious violation of the one-person-one-vote equality requirement. The purpose of this section is to explain mathematically how Congress can bring the House into compliance with the Constitution. The analyses in this section are based on the 2020 population census.

Measuring Inequality

The disparities in congressional district sizes nationwide are illustrated in the chart below, which is based on the apportionment of 435 Representatives that will go into effect in 2023 (with the 118th Congress).1This is based on the 24th apportionment as determined by the United States Census Bureau. See: 2020 Census Apportionment Results.  For each state, the chart indicates the average population of its congressional districts, sorted from the smallest (Montana) to the largest (Delaware).

Average Congressional District Population and Number of Districts per State - 2020 Census

Click on charts to enlarge

Bear in mind that the larger the district, the less political clout its citizens have.

The horizontal red line in the chart above indicates that the average population of all 435 congressional districts is 761 thousand. Note that the number of Representatives granted to each state is also indicated (in parentheses).

Voter inequality can be easily understood in terms of the percentage difference between one state’s average district size and another’s. For example, Montana’s two Congressional districts are the smallest (at 542,704) while Delaware’s single district is the largest (990,837). Therefore, Delaware’s district is 82.6% larger than Montana’s average district size. As it turns out, that means that Montanans have 82.6% more political weight than Delawareans. (See pop-up text box for an explanation of this math.)

Consequently, it could be rightly said that only Montanans will get that “one vote”, while all other Americans get a partial vote. As explained in this section, this political inequality among citizens diminishes as the number of Representatives is increased.

A Larger House Achieves Citizen Equality

Relative to the disparity in congressional district population sizes, two measures can be used to show the degree of inequality that results from any given number of Representatives. The first measure is easy to understand, as it is simply the disparity percentage between the largest and smallest congressional districts for any given apportionment. As explained above, that disparity is 82.6% for a House size of 435 (based on the apportionment which will go into effect in 2023).

The second measure of inequality is relative standard deviation2Also known as the “coefficient of variation”, the relative standard deviation is the ratio of the standard deviation to the population mean, expressed as a percentage.. The preceding chart indicates (in the table) that the relative standard deviation of that apportionment is 2,986% (for all 435 congressional districts). Such a number confirms that the disparity in congressional districts sizes is huge when only 435 Representatives are allowed by Congress.

Because it is a much more accurate measure of dispersion than the disparity percentage, the relative standard deviation will be the primary focus of these analyses.3The Relative Standard Deviation is a better measure because it takes into account the disparity of every instance of a congressional district, whereas the disparity percentage only considers two values: The largest and smallest average district sizes. In addition, the disparity percentage does not reflect the magnitude of the underlying values.  For example, the disparity between 800,000 and 500,000 would be the same percentage as the disparity between 80,000 and 50,000.

However, for those who are not familiar with standard deviation, the disparity percentage is sufficient to understand the problem of political inequality resulting from there being too few Representatives relative to the total population. In any case, these two measures correlate closely, and they both decrease as the number of Representatives is increased, as illustrated in the chart below for all House sizes ranging from 435 to 11,050 (inclusive). The dark (red) graph, plotted against the left Y axis, indicates the relative standard deviation for every House size (horizontal X axis).4The relative standard deviation is calculated from every instance of a congressional district, not just the state average. For example, if there are 5,000 districts, then 5,000 values are used to calculate the relative standard deviation.  And the lighter (blue) graph indicates the disparity percentage between the largest and smallest average district sizes, plotted against the right Y axis.

Nationwide Dispersion in Congressional District Population at Various House Sizes

The chart illustrates that as the number of single-member congressional districts is increased, both measures of political inequality decline significantly. In particular, note how the relative standard deviation asymptotically approaches zero, which is perfect one-person-one-vote equality. However, looking at the table below the chart, at no point in this range does it achieve the degree of equality required by the Constitution’s one-person-one-vote requirement, especially in comparison to the nearly perfect equality established by intrastate congressional districting.5According to the “2010 Redistricting Deviation Table” provided by the National Conference of State Legislatures (NCSL), 45 states had a 0.0% discrepancy between their largest and smallest congressional districts. Of the remaining five states, the largest discrepancy is less than 1%.

For example, even for a House size of 11,036, the maximum allowed by the Constitution for the 2022 census, the relative standard deviation is still 6.8%. Though that would be a vast improvement over the current apportionment of 435, they are still in excess of what is usually required for one-person-one-vote compliance. However, the Constitution’s minimum average district size of 30,000 would override the one-person-one-vote requirement to implement a more equiproportional apportionment.

With respect to House sizes smaller than the Constitution’s upper limit, there is absolutely no constitutional justification for not having a House size that achieves the smallest possible disparity in district sizes. However, if the intended version of Article the first were ratified, that would provide a constitutional basis for not selecting the most equiproportionate apportionment solution. The ratification of this representation amendment would require Congress to select a number of Representatives between Article the first’s minimum (1:50,000) and the maximum specified by the Constitution (1:30,000). Relative to the 2020 population census, this would have required Congress to select a House size between 6,623 and 11,036, as illustrated in the Equitable Representation zone highlighted in the chart above, thereby providing a constitutional override to the one-person-one-vote requirement with respect to House sizes within that range.

With the passage of Article the first (in its intended form), the chart below illustrates a compliant apportionment solution based on an average congressional district size of 49,478, which is achieved with 6,692 Representatives.

Average Congressional District Population and Number of Districts per State - 6,692 Representatives

Note that this apportionment produces congressional districts that are all nearly the exact same size. As a result, the disparity percentage is reduced from 82.6% (in the current apportionment of 435) to less than 4%.  And the relative standard deviation is reduced from 2,956% to 12%. Though this is not perfect equality, it is far more equitable than the flagrant political inequality that Congress imposes on Americans today.

According to their Respective Numbers

Up to this point, the focus of this analysis has been the one-person-one-vote requirement that any legislature’s electoral districts must be equally sized.  However, relative to the House of Representatives, this same requirement is explicitly stated in the Constitution, albeit in a different way, as follows:

“Representatives and direct Taxes6The Constitution originally allocated the federal tax burden to the states on the same basis as representation. However, this provision was overridden by the 16th Amendment in 1913. shall be apportioned among the several States which may be included within this Union, according to their respective Numbers…”

That is, representation is to be apportioned to the states in the same proportion as their respective population totals.  For example, if a state has 1.5% of the population, it should also have 1.5% of the representation in the House.  However, that solution can be suboptimized by two constraints imposed by the Constitution, as follows:

“The number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at Least one Representative”

In the early years of the country, when the populations levels were much smaller, those two constraints prevented a perfectly equitable apportionment. That is, they prevented an apportionment in which each state’s share of representation equaled their share of the total population.7The primary problem is that achieving a truly proportionate allocation of representation would have, mathematically, required an average district size smaller than 30,000. This is what President Washington discovered when he issued the very first presidential veto.  However, as shown in this section, modern population levels have rendered those two constraints nearly inconsequential, making it possible to allocate representation to the states in proportions almost identical to their “respective Numbers”.

For this analysis, the degree of equitability of any apportionment is determined by calculating the ratio of each state’s share of representation to their share of population., which we’ll call the “Rep-Pop ratio”. For example, if a state has 1.5% of the representation and 1.5% of the population, that ratio is 1.5%/1.5%, or 1.0, which is a perfectly equitable apportionment for that particular state.  If a state’s Rep-Pop ratio is greater than one, then excess representation has been allocated to it.  If it is less than one, then it has been effectively shortchanged, with its representational shortfall having been allocated to a different state.

The example provided by Montana and Delaware will be used to illustrate this simple calculation. Starting with Montana, with the smallest Congressional districts, its shares of representation and population is 0.46% and 0.33%, respectively. Therefore, its Rep-Pop ratio is 1.4, indicating that its share of representation is 40% larger than it should be, which is consistent with the previous analysis in this section.

With the largest congressional district in the nation, Delaware’s shares of representation and population is 0.23% and 0.30%, respectively, making its Rep-Pop ratio 0.768.  This means that Delaware has only 77% of the representation that it should in the federal House.

The adjacent table provides this same analysis for all fifty states, starting with each state’s percentage share of the total population based on the 2020 census. Also shown is each state’s percentage share of representation, along with the resulting Rep-Pop ratios, for two different House sizes. The first analysis is the base case of 435 Representatives pursuant to the 24th apportionment which will go into effect in 2023.

The second analysis shown in the table is for the 6,692-Representative scenario suggested above (as it complies with the intended version of Article the first). Note how all of these Rep-Pop ratios are much closer to the 1.0 ratio required by the Constitution.

These Rep-Pop ratios are presented graphically in the chart below. In order to illustrate the malapportionment created by the 435-Representative apportionment, the light green bars indicate which states have excess representation, and the light red bars indicate which states are insufficiently represented (in comparison to the perfect solution of 1.0).

Bar chart of representation to population share ratios.

Also in the chart above are the Rep-Pop ratios for the 6,692-Representative scenario, indicated with the darker green and darker red bars. Because these ratios are much smaller they are barely visible. Note that these ratios (which are provided in the table above) are very close to the perfect ratio of one.

These two examples illustrate the fact that as the size of the House is increased, all of the states’ Rep-Pop ratios approach one. Accordingly, the disparity in those values from state to state diminishes rapidly. This can be determined by calculating, for each apportionment, the standard deviation of the Rep-Pop ratios for all 50 states. As shown in the chart below, as the number of Representatives is increased, the standard deviation from unity (1.0) decreases significantly.

Standard Deviation of Rep-Pop Ratios for all House sizes between 435 and 11,050.

As an example, whereas the standard deviation for the 6,692-Representative scenario is .006, the standard deviation of the 435-Representative apportionment 18 times larger, at .115.

Note how the chart above correlates with the relative standard deviation of congressional district sizes nationwide in the preceding chart.8At 0.93, the coefficient of correlation is nearly perfect. That is because these two analyses are actually two sides of the same coin.

The first analysis is based on the Supreme Court rulings which instituted one person one vote equality for legislative bodies. Therefore, it focused on the variation in congressional district population sizes across the nation. The relative standard deviations resulting from the various House sizes show how this disparity is reduced significantly as the number of Representatives is increased.

The second analysis, which is based on the Constitution’s requirement for proportional allocation of representation, focuses on the states’ Rep-Pop ratios.  And as the number of congressional districts is increased, not only do these ratios approach the perfect ratio of 1.0, but their dispersion in values decreases significantly (as indicated by their standard deviation).

The reason that these analyses are two sides of the same coin is that those states with the smaller congressional districts are the same ones which have high Rep-Pop ratios. And conversely, the states with the larger congressional districts have smaller Rep-Pop ratios.  For example, as shown above, Montanans have 82.6% more political weight than Delawareans because the latter’s congressional district is 82.6% larger than Montana’s (as a result of the 24th apportionment).  The same result is arrived at by comparing these states’ Rep-Pop ratios as follows:

This ratio, 1.826, equates to the 82.6% political advantage held by Montana vis-à-vis Delaware.

A Constitutional Apportionment

If one-person-one-vote equality were the basis for apportioning the House, as required by the Constitution, then Congress would choose the number of Representatives that minimizes political inequality among the citizens of the different states.  Relative to the 2020 population census, the number of Representatives which appears to produce the most equitable apportionment is 10,295 (with an average district size of 32,162).  This House size results in the lowest relative standard deviation in congressional district population sizes. Any other House size that does not produce a comparably equitable apportionment is indisputably unconstitutional.

However, the passage of the representation amendment intended by the first Congress, Article the first, would provide constitutional authority for Congress to select any number of Representatives between a minimum of 1:50,000 (6,623) and a maximum of 1:30,000 (11,036) regardless of the resulting variation in congressional district sizes.  This would allow the House of Representatives to avoid the requirement for near perfect equality that is currently imposed on the creation of intrastate congressional districts, as well as the creation of virtually every other electoral district in the country. Otherwise, absent such a constitutional provision, why should state and local legislative bodies be required to meet a standard that the most important legislature in the country chooses to egregiously violate?

© Thirty-Thousand.org [Article published 03/01/22, and updated 04/24/22]

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  • 1
    This is based on the 24th apportionment as determined by the United States Census Bureau. See: 2020 Census Apportionment Results.
  • 2
    Also known as the “coefficient of variation”, the relative standard deviation is the ratio of the standard deviation to the population mean, expressed as a percentage.
  • 3
    The Relative Standard Deviation is a better measure because it takes into account the disparity of every instance of a congressional district, whereas the disparity percentage only considers two values: The largest and smallest average district sizes. In addition, the disparity percentage does not reflect the magnitude of the underlying values.  For example, the disparity between 800,000 and 500,000 would be the same percentage as the disparity between 80,000 and 50,000.
  • 4
    The relative standard deviation is calculated from every instance of a congressional district, not just the state average. For example, if there are 5,000 districts, then 5,000 values are used to calculate the relative standard deviation.
  • 5
    According to the “2010 Redistricting Deviation Table” provided by the National Conference of State Legislatures (NCSL), 45 states had a 0.0% discrepancy between their largest and smallest congressional districts. Of the remaining five states, the largest discrepancy is less than 1%.
  • 6
    The Constitution originally allocated the federal tax burden to the states on the same basis as representation. However, this provision was overridden by the 16th Amendment in 1913.
  • 7
    The primary problem is that achieving a truly proportionate allocation of representation would have, mathematically, required an average district size smaller than 30,000.
  • 8
    At 0.93, the coefficient of correlation is nearly perfect.