"_____ is the 3,180th most popular last name (surname) in the United
States; frequency is 0.004%; percentile is 57.748" is a typical result
for uncommon last names (try out your own name for fun).  I cannot
understand the precise use of  "percentile".  For the name given
above, I know that about 10,000-100,000 of these people live in the
US, so the frequency seems to reflect the ratio of the number of
citizens with this name divided by the number of citizens, but to what
does "percentile" refer?  For many names with percentiles less than
0.000%, the percentile is still in the mid-range.  Pleas provide an
example.

The answer to your question is as follows:

The percentile of 57.748 is the cumulative % of the population covered
by names ranking 1 through 3180.  As in, 57.748% of the population
have that last name or one more common.  The reason why the number
increases as the frequency of actual occurrance decreases is because
they are ranking the names by greatest frequency of occurrance to
least.  The first ranked name only contains a small percent, the
absolute rarest name on the census list (they only list through the
88799th most common) has a cumulative frequency of 90.483%.

For example, % showing frequency of the #1 ranked last name (Smith) is
1.006%, the cumulative % for the #1 ranked name is 1.006%.  The
frequency of the 2nd most common last name (Johnson) is .810%, making
the cumulative % of names ranked 1 through 2 1.816%.

I used http://www.census.gov/genealogy/names/names_files.html and
http://www.census.gov/genealogy/names/dist.all.last as my sources.

Thank you for the answer!  I understand the use of percentile in this
context, but it was not obvious to me.