"It depends" is not the answer most want to hear to a question, but in this case you do have the answer. It depends on how big you are, how big your tub is, what's the flow rate on your shower head, and how long you like to enjoy your shower time. With these variables we answer the age old question as well as show you some bathroom tech that can help cut down on your water bill.
Only you can answer the aforementioned questions, and once we show you the steps you can calculate the answers for you and your bathroom. For the sake of discussion we will use a few fixed variables from supposed averages to figure out which uses more water - taking a bath or shower.
For bathtubs, it's generally a simple matter of how big your tub is. A search at Sears, Home Depot, and other bath tub sellers comes up with such a huge range of tub sizes, with most settling in the 50-80 gallon range.
You'll also need to think about displacement, because as you're sitting in the tub the space that your body occupies doesn't need to fill with water. Answers.com says
The displacement in gallons would be about equal to the weight of the individual since the density of a average body is about equal to the density of water. Larger bodies float and smaller bodies sink due to fat being less dense than water and muscle. Given that an average body weighs 155 lbs, we divide by 8.34 lbs/gal to get about 18.5 gallons water.
Hopefully you're not always fully submerged the entire time you're sitting in the bathtub, but even given the full 8+ gallons of water displacement from a 50 gallon bathtub, you're using 40+ gallons of water.
Your showerhead is rated for gallons per minute (GPM) as a way of determining how much water flows through it. Federal law mandates that all showerheads manufactured and sold after January 1, 1994, must use no more than 2.5 GPM. If the average shower is under 10 minutes long, simply multiply the GPM by the length of the shower. For a 2.5 GPM shower head over an average 10 minute shower length, that's 25 gallons of water used.
Install an In-Home Water Use Monitoring System: If you want to get hi-tech about monitoring how much water is being used by both shower and bath takers in your household (and water use everywhere else in your home), you can invest in something like this ORION in-Home Display system, though what we really want to see is a more user-friendly app based water and energy monitoring systems accessible at a glance from your smart phone or tablet.
Your mileage may vary depending on your habits, of course. But habits can be changed, and there are several things you can do to save water when showering. We found several tech tips and gadgets from the Unplggd archives to help you save on your water bill.
Install a low-flow shower head
Install a shower timer
Monitor your usage