Huntington-Hill Method Method of Equal Proportions Joseph A. Hill, Chief statistician, Bureau of the Census Edward V. Huntington, Professor of Mathematics, Harvard

Some History Developed around 1911 Endorsed by a National Academy of Sciences Panel in 1929 Signed into law as our apportionment method in 1941 when the House size was set at 435.

Two Questions to Answer Why was the Huntington-Hill method chosen as our apportionment method? How does it work? We will answer the second question first. (To make our life easier, we will abbreviate the method as HH!)

Divisor methods and Means The HH method is a divisor method. The difference between it and other divisor methods is how the quotient for each state is rounded. To understand this rounding method, we take a side trip and talk about three different types of means and how they are used to round in 3 different divisor methods.

Arithmetic Mean The arithmetic mean(AM) of two numbers is their usual ‘average’: AM(12,13) = ( )/2 = 12.5 Webster’s method uses this number as the cutoff for rounding (round down if the quotient is less than this number.)

Geometric Mean The geometric mean(GM) of two numbers is the square root of their product: GM(12,13) = √(12∙13) = Huntington-Hill uses this number as the cutoff for rounding (rounding down if the quotient is less than this number.)

Harmonic Mean The harmonic mean(HM) of two numbers is twice the product of the two numbers divided by their sum: HM(12, 13) = 2∙12∙13/( ) = Dean’s Method uses this number as the cutoff for rounding (rounding down if the quotient is less than this number.)

How do the means compare? Take four number pairs (4 and 5, 6 and 7, 8 and 9, 10 and 11) and calculate each of the three means for that pair. Then compare the sizes of the means and write down your results (GM<HM<AM, say). What do you notice? Test your conjecture with another pair like 37 and 38.

An example Do HH and Dean for a divisor of 270 SchoolPopulation East1500 West1000 North9000 South6000 Central200