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We present a technique to explore the radio sky into the nanoJansky regime byemploying image stacking using the FIRST survey. We first discuss thenon-intuitive relationship between the mean and median values of a distributionthat is dominated by noise, followed by an analysis of the systematic effectspresent in FIRST's 20cm VLA snapshot images. Image stacking allows us torecover the properties of source populations with fluxes a factor of 30 or morebelow 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 quasarsfrom the SDSS DR3 catalog. There is a tight correlation between optical andradio luminosity, with the radio luminosity increasing as the 0.85 power ofoptical luminosity. This implies declining radio-loudness with opticalluminosity: the most luminous objects (M=-28.5) have average radio-to-opticalratios 3 times lower than the least luminous objects (M=-20). There is also astriking correlation between optical color and radio loudness: quasars that areeither redder or bluer than the norm are brighter radio sources, with objects0.8 magnitudes redder than the SDSS composite spectrum having radio-loudnessratios that are higher by a factor of 10. We explore the longstanding questionof whether a radio-loud/radio-quiet dichotomy exists in quasars, finding thatoptical selection effects probably dominate the distribution function of radioloudness, which has at most a modest (~20%) inflection between the radio-loudand radio-quiet ends of the distribution. We also find, surprisingly, thatbroad absorption line quasars have higher mean radio flux densities, with thegreatest disparity arising in the rare low-ionization BAL subclass.Comment: 17 pages, 19 figures, accepted by ApJ; corrected error in absolute  magnitude

Signals from the Noise: Image Stacking for Quasars in the FIRST Survey