Below is a chart showing the percentage margin of majority in each chamber since 1869 (the point at which the Confederate states had rejoined the Congress). So if Republicans hold 53 seats in a 100-person Senate to the Democrats' 45, they have an eight percent margin of majority. (We excluded caucusing independents and, in the case of resignations and deaths, used the point at which the GOP had the highest membership.) And if they hold 243 seats in the House to the Democrats' 192, that's an 11.7 percentage point advantage -- or a 19.7 percentage point combined majority in the Senate and House.
That's slightly higher than the Republican dominance in the 80th Congress of 1947 (which had a smaller Senate). The most recent larger margin was in the 71st Congress, as mentioned above. (As our Reid Wilson noted last week, 2015 could also see the GOP controlling the most state legislatures since 1920.)
One of the more remarkable aspects of the chart above is the reminder that single-party dominance was the norm for much of the 20th century, with Democrats holding both chambers for Congress after Congress even as the White House changed hands. The recent back-and-forth between the parties is the exception, not the norm.