You can buy weigh scales from tanita that have Bluetooth links to your home pc. They are pretty good scales, have a set myself. Then its just a little bit of programming to do your rolling average filter.
If you want to make your own equivalent, then you would need to build an instrumentation amplifier to amplify the strain guage bridge in the scales, this would give you an analog value for the weight on the scales, which you could read with an arduino. Then you would have to calibrate the scales for weight, figure out how to power the arduino in the bathroom, and how to communicate out the values.
Personally I would just buy the Bluetooth scales and get on with more fun projects