I would say it probably has to be off quite a while. Some tank heat loss tables I have used for other calculations say a steel tank with 1" of insulation, filled with 120F water in 60F still air will loose about 25 BTU per SQ/FT per hour. A 50 gallon water tank has about 21.3 SQ/FT of surface area, so that is about 532 BTU of heat loss per hour. It takes 4000 BTU of loss to drop the tank temp on 400# of water 10F. So the tank would probably only be down about 10F in 8 hours. An electric heating element delivers about 3413 BTU per KW per hour, so a standard 4.5KW heating element would heat at a rate of 15,359 BTU, which would put the tank back up to 120F in about 15 minutes, at a cost of around 1 KW hour(multiply by your local power rate). If you use hot water in the morning and the tank warms to shutoff temp, it probably dosn't run again all day unless it is in a drafty unheated basement space. Most tanks also have more than 1 inch of insulation around them, and more is better in this case.
It should take about 1.56 hours for the 4500W element to heat a tank from 60F to 120F. That is a little over 7KW hours of energy(again multiply by your local rate). If your power rate is say 10 cents per KW/HR, then it would take 70cents to heat that tank of water from 60F to 120F. If not being used, and the tank only re-heats say 2.5 times a day at 10 cents per re-heat, you would have to turn it off for about 3 days to equal that 70cents to fully re-heat a tank that has cooled to room temperature. Under the above conditions, I would guess that any day after 3 days, you are saving about 25 cents per day... These numbers are very rough, so YMMV.
