There are several ways to measure the discrepancy between the assigned values and quotas. I shall calculate two of these for Rhode Island and Connecticut.
Exercise: For which state is Ai-Qi greatest? least? What
does this represent?
For which state is Ai/Pi (or Ai/Qi) greatest? least? What does this represent?
For which state is Pi/Ai (or Qi/Ai) greatest? least? What does this represent?
Using Ai/Qi or Qi/Ai is nice because unfairness is measured from the ratio 1.
The first apportionment method employed (1791-1850), called Greatest Divisors or rejectected fractions and propounded by Thomas Jefferson, maximized the minimum district size. Since the district size is Pi/Ai, the minimum district size for the original apportionment is 23614 in Georgia. Since a small district size means more power per voter, maximizing the minimum district size entails minimizing the extent to which anyone has too much power. Note that Pi/Ai = (Qi/Ai)(P/H), hence the minimum of the ratio Qi/Ai is maximized or the maximum of Ai/Qi is minimized (overrepresentation is minimized)
The method used from 1850-1910, called largest fractions or Hare quota and propounded by Alexander Hamilton (but vetoed by George Washington), minimizes the sum over all states of |Ai - Qi|. For the original apportionment this is readily calculated as 11.59.
The present method (1910-date), called equal proportions or method of the geometric mean, minimizes |log((Ai/Pi)/(Aj/Pj))| over all i,j. For the original apportionment the maximum of this ratio is equal to .52 for the contrast of Georgia and North Carolina. This method minimizes the the disparity of representatives per voter between the states, measured as a ratio.
Both the method of greatest divisors and method of the geometric mean allows a discrepancy greater than 1 in magnitude between the Ai and Qi. The metod of largest fractions allows the "Alabama paradox" by which increasing the size of the House can result in a state losing a representative.
Competency: For the apportionment NH-2, MA-9, RI-1, CT-4, NY-6, NJ-3, PA-8,DE-1, MD-5, VA-14, NC-7, SC-4, GA-1;: calculate Qi-Ai for each state, Qi/Ai foreach state, Pi/Ai for each state (compare the minimum to the value calculated with the original apportionment), the sum over all states of |Ai - Qi| (compare this to the value calculated above for the original apportionment), and the maximum of |log((Ai/Pi)/(Aj/Pj))| over all i,j (compare this to the value for the original apportionment). (This is the apportionment for the method of greatest divisors.
I am really more interested in the differences Ai-Qi and the ratios Ai/Qi because these directly measure over or under representation.