Okay, because I have nothing better to do on a Friday night (i.e., my carboys are all currently fermenting), I've decided to fiddle a little bit with math.
I know that 0% alcohol by weight (ABW) is also 0% alcohol by volume (ABV), and I know that 100% ABW is 100% ABV. I also know that, in between those two extremes, the relationship is not linear (e.g., 3.2% ABW is approximately 4.0% ABV). However, any web-site calculator that I see pretty much performs the calculation as linear.
I've been fiddling with my TI-89 from high school, and I think that I have it figured out. Assuming ethyl alcohol has a density of .789 kg/l and water has a density of 1 kg/l and assuming that water and alcohol are the only substances in an alcoholic beverage (which I know isn't true), the formula is:
ABV = ABW ÷ (.211 · ABW + .789)
The problem is that I can't confirm my formula. I'm searching Google and can't find much on the topic. Well… I can find plenty of calculators, but they all seem to be using a linear formula.
So… is this formula correct?
EDIT: So, I've done a little more tweaking on this (after I already got the accepted answer). I already made the assumption that the density of water is 1 kg/l. The density that I used for ethyl alcohol was for 20°C. At that same temperature, water has a density of approximately .99823 kg/l and not 1 kg/l. Using this, I arrived at a different formula that's probably more accurate (at 20°C, that is):
ABV = (99823 · ABW) ÷ (20923 · ABW + 78945)
EDIT: Please see AlkonMikko's comment.