Congressional Districts Over the Years

Redistricting season is here, now that the U.S. Bureau of the Census has released the 2020 census apportionment data. As mapmakers clean their quills and the political parties sharpen their arguments, it would be interesting to see how Ohio’s districts have shaped up over the years.

Ani Ruhil

Article 1, Section 2, of the United States Constitution states:

“Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers…The actual Enumeration shall be made within three Years after the first Meeting of the Congress of the United States, and within every subsequent Term of ten Years, in such Manner as they shall by Law direct.”

Therein lies the primary mandate of the U.S. census, apportionment of the House of Representatives. Since that first census in 1790, five methods of apportionment have been used. The current method used, the Method of Equal Proportions, was adopted by congress in 1941 following the census of 1940. This method assigns seats in the House of Representatives according to a “priority” value. The priority value is determined by multiplying the population of a state by a “multiplier.”

Each of the 50 states is given one seat out of the current total of 435. The next, or 51st seat, goes to the state with the highest priority value and becomes that state’s second seat. This continues until all 435 seats have been assigned to a state. This is how it is done. (Source:

Some interesting developments over the years:

Speaking of various methods, what options have been in play?

(a) The Hamilton-Vinton Method

Take the total population of the 50 states and divide by the total number of seats to be apportioned. This will give you the divisor. Divide each state’s population by the estimated divisor. This gve you the first allocation of seats, but will result in surplus seats because the decimal has been ignored. The state with the largest leftover decimal gets the first surplus seat, then the state with the next largest leftover decimal, and so on until the total number of available seats have been allocated. If you sum First Allocation you get 410, leaving us with 25 surplus seats. Thus, Alaska gets a seat because it has the largest leftover decimal value, followed by Vermont, then Wyoming, and so on.

# A tibble: 1 x 2
   totalpop divisor
      <dbl>   <dbl>
1 331108434 761169.

The Jefferson Method

The Jefferson Method avoids the problem of an apportionment resulting in a surplus or a deficit of House seats by finding and using a divisor that will result in the correct number of seats being apportioned.

The Webster Method

The Webster Method modifies the Hamilton/Vinton method by awarding an extra seat to any state with a quotient that has a fraction above 0.5. While the total number of seats to be apportioned is set to calculate the divisor, this total can be increased if a large number of state have fractions exceeding 0.5.

The Huntington-Hill Method

The Huntington-Hill Method further modifies the Webster method by rounding at the geometric mean (the square-root of the product of two numbers). The geometric mean will require estimating two numbers – the Lower Quotient (with no rounding), and the Upper Quotient (with rounding up to the nearest integer). States with a quotient exceeding the geometric mean receive an additional seat, and this method will usually yield the desired number of total seats.

In the Equal Proportions Method rhe key parameters are \(p\) (the state’s population size) and \(n\) (the number of seats if the state gained a seat from the initial allotment of 1 seat). The multiplier is calculated as \(\dfrac{1}{\sqrt{(n(n-1))}}\), which leads to the following multiplier values per additional seat:

Now that the multipliers are ready, the next step would be to calculate \(multiplier * p\) for each state. Stack the resulting priority values in descending order, assigning one seat for each of the 435 rows of states that show up. Each time a state shows up in the first 435 rows of data, it gets an additional seat. For 2020 this will result in 44 states receiving additional seats beyond the first seat automatically assigned to each of the fifty states. The remaining six seats will end up with a total of one seat, and these states will be Alaska, Delaware, North Dakota, South Dakota, Vermont, and Wyoming.

Now, if we want to zoom in on Ohio, we could look at change in our Congressional Districts since statehood. As we draw these, do remember that since statehood and up through the 1805-1813 Congresses we had one at-large Congressperson. Another at-large representative in the 1913-1915 Congress represented the \(22^{nd}\) district starting in the 1915-1917 Congress. Two at-large seats were added in the 1933-1943 years, but only one was retained over the 1943-1953 period, eventually flipping into the \(23^{rd}\) district. Another at-large seat arose in the 1963-1967 years, flipping into the \(24^{th}\) district until 1973, when the seat was lost. Two more seats were lost after the 1981-1983 Congress.

Historical data snippets are available here and are definitely worth browsing.


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For attribution, please cite this work as

Ruhil (2021, April 27). From an Attican Hollow ...: Congressional Districts Over the Years. Retrieved from

BibTeX citation

  author = {Ruhil, Ani},
  title = {From an Attican Hollow ...: Congressional Districts Over the Years},
  url = {},
  year = {2021}