the flaw of average drunks

That was one of the lessons of an intensive December course. Basically, the expected value of the average isn't the same as the average of the expected values. A much nicer way to put it follows...

It's the story of a drunk who staggers back and forth between two traffic lanes. The state of the drunk at his average position is alive. But the average state of the drunk is still dead.(Savage & Van Allen, 2002)