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Minimizing ______ prioritizes each state ’ s number of representation to be as close to the national rate as possible versus each state ’ s representative of representatives to be closest to its quota. Dean ’ s method was never used. However, it is im portant to mention that Hill appreciated Dean ’ s views on equity in representation, but he proposed a slight modification. While Dean minimized the absolute difference between the representations ______ , Hill felt it was more appropriate to mini mize the relative difference. That is, minimize ______________. This ensures that if a seat were transferred from one state to another, the minimum relative deviation from the national representation of the pair would increase. While the absolute and relative differences will move in tandem, they can differ. Two examples help us understand the dif ference between these two measures. Hill used a hypothetical example. Consider two pairs of states. In the first pair, one state has 100,000 people per representative and the other state has 50,000 people. In the second pair, one state has 75,000 people and the other 25,000 peo ple. Both pairs exhibit the same absolute differ ence. This Dean would say the underrepresented states share the same measure of inequality. How ever, the underrepresented state in the second pair has a relative difference of

quota was 3.51. If these two states were to split 19 representatives, should they be split 15 - 4 or 16 - 3? As shown in Table 4, when Massachusetts has 15 seats, one Massachusetts representative represents 36,273 more people than one Florida representa tive. If Massachusetts increases to 16, now Florida is underrepresented with an absolute difference of 40,472 people per representative. Dean chooses the smaller of these two values and gives Massachu setts 15 and Florida 4. Hill, however, looks at the relative difference: when Massachusetts has 15 seats, one Massachusetts representative represents 19.3% more people than one of Florida ’ s repre sentatives. Giving one of Florida ’ s seats to Massa chusetts makes Florida underrepresented, but by only 19.2%. Thus, Hill would disagree with Dean and say it is preferable to give Massachusetts the 16 seats. Hill ’ s presentation to Congress contained an error in his algorithm, but Edward V. Huntington, Hill ’ s Harvard classmate who became a Harvard profes sor and statistical consultant to the military in World War I, caught this error, and so the correct ed method is known as the Huntington - Hill meth od. The Hill - Huntington method is also known as the method of equal proportions. Whereas Dean ’ s method uses harmonic mean rounding, the Huntington - Hill method is achieved by rounding the quota with respect to the geomet ric mean of the integers above and below the quo ta. That is, _______ . This number is always great er than the harmonic means of n and n +1 but less than their arithmetic mean. Continuing our previ ous example of rounding 4.48, the geometric mean of 4 and 5 is _________ , and so the Huntington Hill method would round up like Dean ’ s method on the other hand, Webster ’ s Method would round down.

This is worse than the underrepresented state in the first pair, which has a difference of only

Balinski & Young (2001) compare two states from the 1910 Census with N = 425 representatives. Massachusetts had a quota of 15.71 and Florida ’ s

Table 4: Balinski & Young ’ s example comparing the methods of Dean and Huntington - Hill.

Virginia Mathematics Teacher vol. 48, no. 1

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