  We present a technique to explore the radio sky into the nanoJansky regime by
employing image stacking using the FIRST survey. We first discuss the
non-intuitive relationship between the mean and median values of a distribution
that is dominated by noise, followed by an analysis of the systematic effects
present in FIRST's 20cm VLA snapshot images. Image stacking allows us to
recover the properties of source populations with fluxes a factor of 30 or more
below the rms noise level. Mean estimates of radio flux density, luminosity,
etc., are derivable for any source class having arcsecond positional accuracy.
  We use this technique to compute the mean radio properties for 41,295 quasars
from the SDSS DR3 catalog. There is a tight correlation between optical and
radio luminosity, with the radio luminosity increasing as the 0.85 power of
optical luminosity. This implies declining radio-loudness with optical
luminosity: the most luminous objects (M=-28.5) have average radio-to-optical
ratios 3 times lower than the least luminous objects (M=-20). There is also a
striking correlation between optical color and radio loudness: quasars that are
either redder or bluer than the norm are brighter radio sources, with objects
0.8 magnitudes redder than the SDSS composite spectrum having radio-loudness
ratios that are higher by a factor of 10. We explore the longstanding question
of whether a radio-loud/radio-quiet dichotomy exists in quasars, finding that
optical selection effects probably dominate the distribution function of radio
loudness, which has at most a modest (~20%) inflection between the radio-loud
and radio-quiet ends of the distribution. We also find, surprisingly, that
broad absorption line quasars have higher mean radio flux densities, with the
greatest disparity arising in the rare low-ionization BAL subclass.
