How is each states representation the House of Representatives determined?

Answer 1

Every ten years, after each U. S. census is completed, the 435 seats in the House of Representatives are reapportioned to the states based on each state's relative population. The only time the number of House seats deviates from 435 is between the time that the number of states changes and the time of the next reapportionment.

First, one House seat is assigned to each state, since that is the minimum that each state must have. Then for each state they calculate the state's population divided by the square root of the product of the number of House seats assigned to that state so far and that number plus 1. The states are listed in descending order according to those numbers. Since at this point each state has been assigned one seat so far, the list is initially in descending order according to population. The next House seat is assigned to the state at the top of the list, that state's number is recalculated, and the state is moved down list according to its new number, so the states are again in descending order by their calculated number. The next House seat is assigned to the state that had risen to the top of the list, its new number is calculated, and it moves down so it's in order by its new number. The assignment continues like this until all of the House seats have been assigned.

While a state has one seat assigned to it, its place in the list is determined by its population divided by 1.414 (the square root of 2). As the number of seats per state approaches infinity, the number by which its population is divided approaches the number of seats assigned to the state so far plus one half.

The object of the reapportionment is to get as close as mathematically possible to every Representative representing the same number of residents, and this method was found to work best to that end. So if you calculate each state's ratio of number of people per Representative, the standard deviation of all those ratios should be as low as possible. To give a simplified example, let's say that the country has four states, and 10% of the country's population lives in State A, 20% lives in State B, 30% lives in State C, and 40% lives in State D. When the apportionment is done, 10% of the House seats should have been assigned to State A, 20% to State B, 30% to State C and 40% to State D.