In general terms the question I'm wondering about is, what is the "average" answer in the US to "how long have my ancestors lived in the US"
This is one way of making it precise. For each person living in the US, there is a year at which point 50% of their ancestors were living in America, call it the year of ancestral immigration (YAI). If you are a first generation, your (YAI) is the year you immigrated. If one of your parents immigrated and the other was born in the US then your YAI is the year the immigrant parent arrived (50%). This can go as far back as necessary.
There may be better ways of making the general sentiment precise. Other relevant information would also be appreciated. For instance, perhaps you could say that "50% of people living in the US are descended exclusively from ancestors who arrived after 1897" etc.
I'm very curious about this, so your answers are much appreciated.
Interesting statistic. I don't think it is available.
It might be possible to model it though. The % of the population not born in the US is reported in the US census and available quite a way back. From memory, it is about 12% today and has never been higher than 15% (which point was sometime back in the early 1900s) (It has had long periods of being considerably lower between then and now). (You would need to check these statistics. They are available on line, but I am just going by memory.)
With the percent of the population born abroad every 10 years, and the population of the country from the same censuses, with some assumptions, it should be possible to model your statistic.