My lightest bass is 6.6 pounds. I build a lot of light basses, and I know what it takes to make a bass light - every piece has to be light to get under 7.5 on a 4 or under 8 on a 5 string. My 6.6 pounder has an insanely light piece of swamp ash for a body, a body shape that's smaller then a jazz, only one pickup, a roasted neck with graphite reinformcement rods, all aluminum hardware (bridge, tuners, neck plate, even the 2 knobs it has are aluminum). It doesn't thave a pickguard (thats about 60 grams or .15 pounds). I highly doubt that any Fender bass with the original clunky steel hardware (tuners, neck plate, bridge, and control plate) weighs less than 6 pounds. Especially a Jazz - bigger body, two pickups, etc.
My 6.6 pound figure is from two scales I have, which weigh to the gram and which always agree with each other within a gram. Technically, that bass is 2984 grams (6.58 pounds), but if you get to that level of precision, and have decent scales, you'll find things like basses weigh 20 or 30 grams less in the winter (in my climate) because they're drier. And, of course, the gauge of strings you put on will change the weight a bit.
A bathroom scale isn't even close to good enough to weigh a bass accurately, but you can get postal or kitchen scales that work very well for that weight range for reasonable prices - I have one of each.