~12kWh/month = ~400 watts/day, or about 16.6 watts/hour if constant over 24 hours.
Ohm's law tells us that, at 120 V, you would require about 0.14 amps to produce 17 watts of power. So, you're looking at a phantom draw about on par with leaving a cell phone charger plugged in 24/7.
Point being, if you're serious about saving money on your electric bill, buy a Kill-a-watt meter and take a close look at the devices you leave always plugged in. Many common devices draw power even when they are not "on."
[[Also, I don't entirely buy their "wind blowing spins the fan" story; more likely, there's a transformer in the AC for a control circuit that is drawing power.)
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