What's the minimum amount of electricity used by an appliance?
To test, plug a non-electronic lamp in, and turn the lamp off.  Your P3 should show zero watts, or it is clearly defective.  

Many electronic devices have "parasitic" or "phantom" or "vampire" power usage: I have a toothbrush charger that shows about four watts, 24 hours a day, 7 days a week.  There's a radio that when off shows about ten watts.  And several Apple chargers that show a big zero when charging is done, due to better technology.

Note with the P3 you'll have to measure over a period of a few days, to accurately measure devices that draw less than one watt.

Here's an example study on the topic:
http://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?article=1046&amp;context=eesp
See also:
How do you determine how much electricity is used by a device?