The average wind does puzzle me a little more, to be honest, since they give every next plane that lands, a different wind to the one before.
The average is a two minute rolling value. So, let's say that aircraft are landing one minute apart, each reported wind is the average over the previous two minutes and will probably be different from that preceding it. Just to make things more interesting, the 'instant' wind that Tight Slot refers to is also actually an average - for most systems I think it's a 5 second rolling average.
The wind reported by ATC has always been averaged. In the old days by an element of hysteresis in the display system and often by the controller making some professional judgement, now we just try to do it more systematically and consistently with electronics.