The perils of calculating an Average of Averages

I've seen questions around issues calculating averages come up a few times in various forums and it came up again last week and I feel that there is some benefit in walking through the details of this issue. For many of you the following will be nothing new, but I'm hoping that this may serve as a reference that you can point to when you get requests for this sort of calculation. The core issue here is really a fundamental mathematical one. Personally I see it surfacing most often in DAX and MDX as those are languages that I spend a lot of time with, but also because of their multi-dimensional natures you need to be able to write generic calculations that will work regardless of how the end users slice and dice the data.