It’s about geometry, rather than stats, but it works for me.
Think of it this way:
- If you hold a cane in the air, you can move it in any direction, twirl it, and so on. Its motion isn't constrained at all. That is, the top of the cane can move freely in three dimensions.
- If you put (and keep) one end on the ground, now its motion is constrained: you can't lift it, or rotate it... although you can swing the top around in a variety of different arcs. That is, the top of the cane can move freely in two dimensions.
- If you connect the tops of two canes together and place the other ends on the ground, you can still move the tops, but only along a single (straight) arc, back and forth. That is, the tops of the canes can move freely in one dimension.
- If you try the same trick with three canes, now you can't move the tops at all. This is basically what's happening with a three- legged stool. The tops of the cans can move in zero dimensions... which is to say, they can't.
Each time you add a cane, you remove one dimension in which the top can move freely - that is, each new cane removes one 'degree of freedom'.