Prodded by a comment from Steve Sailer in my Immigration Footnote post, I wanted to look at the number of voters per Congressional district after the federal government was established. I wasn't able to find any pre-20th century election results on-line. The earliest results I could find were from 1920, as kept by the Clerk of the House. In 1920, the population of the US was 106 million and the House had reached its modern size of 435 Representatives. On average, each Representative had 240,000 constituents. In the 1913 reapportionment, each District was meant to have 212,000 constituents.
As one would expect, the number of voters was lowest in the South. I assume that this was due mostly to limits on the franchise, although it could also be that there was no real contest since only Democrats won in the South. In any event, the lowest number of voters came in the Third District of Louisiana, which elected WP Martin (Dem) unopposed (all of Louisiana's Representatives were elected unopposed) with 4201 votes. That is approximately 2% of his constituents.
At the other extreme, Representative Cleveland A. Newton (Rep) was elected from Missouri's Tenth District with 122,100 votes out of 199,729 cast.
Two other 1920 races are worth mentioning. In California's Tenth District, Republican Henry Z. Osborne beat Socialist Upton Sinclair 97,469 to 20,439.
Also, from 1913 through 1923, Pennsylvania had a number of at-large Congressional districts along with its 32 geographically defined districts. I believe, although I can't quite prove, that the top four at-large vote getters, Republicans who got between 1,140,836 and 1,108,538 votes, were elected to Congress from the 33rd through 36th Districts. Frankly, I did not know that a mixture of at-large and geographic districts was possible (it was apparently the result of a refusal or inability to redistrict after the 1910 census). However, this year Democrats are trying to expand the House by two seats, one to be elected from the District of Columbia and the other to be a fourth District in Utah to be elected at-large until the next redistricting in 2012. The bill has passed the House but without sufficient support to overturn a veto. The bill is still in committee in the Senate, but opponents claim that they can successfully filibuster it. The Bush Administration has suggested (but not promised) that it will veto the bill if it reaches the President's desk.