Congressional apportionment

The number of representatives in the House to which each state is entitled is supposed to be proportional to the population of each state; however, it is required to be an integer for each state as well.  For the last 8 House reapportionment cycles, the mechanism used has been one that gives an apportionment that has the smallest possible standard deviation of the logarithm of the population per district; for each state, figure out how many people there are per congressional district and find the logarithm, then look at the standard deviation over the 50 states.  If you weren't forced to use integers, you'd like to make it 0; instead, you make it as small as possible, conditional on there being 435 districts allocated as integers.

The 2000 apportionment created perhaps even more fuss than usual because Utah and North Carolina were so close to each other in terms of which was in line for the 435th representative and which the 436th.  This is related to the fact that the standard deviation of the log district size was quite high relative to what it would have been if there had only been 434; if North Carolina hadn't received the additional district, it would have been more in line with the other states, and adding one to Utah after that would have improved things slightly.  It might have made sense, on these grounds, to shrink the House a little bit to make district sizes a bit more uniform.

Now, if there were one congressman per person, the restriction to integers wouldn't be an issue; all districts would be the same size, and the standard deviation would be 0.  Even if there were say 3000 congressmen, it should be intuitively clear that the stage in which one approximates a perfectly proportionate definition with integers is going to involve much smaller (proportional) changes than for 435, which is to say that there is, on a large scale, going to be a general downward drift, with the standard deviation going approximately as the inverse of the number of congressional districts.  Adding representatives is obviously costly at some point, primarily in the functioning of the legislative body (pecuniary staff costs and so on are presumably dwarfed by this); if you're simply going for the minimum global value, but you want a sane minimum, you have to add a term that grows in the number of representatives.  There's a reason to make this inversely proportional to the number of states, i.e. to add a cost per (congressmen per state) rather than per congressman per se; the plot below shows the standard deviation of log district size plus 1% times the number of congressmen per state, viz. the standard deviation plus House size divided by 5,000.  Aside from seeming like nice numbers to a species with ten fingers, it happens to suggest a House size near the size of the actual House of Representatives.  The exact minimum is 413, but any number between 410 and 430 is fairly close.

As it happens, the optimum for the 2010 allocation is a bit bigger, but similar; the minimum is at 424, but anything from 421 to 429 is close.  If the size of the House were reduced to 424, California and Texas would each lose 2 of the 11 representatives; Florida, Georgia, Minnesota, New York, Pennsylvania, South Carolina, and Washington would lose the other 7.