Originally posted by Riger82:

In one of the videos I saw, I watched them get a level up for jumping. Does increasing the level of a skill that uses stamina decrease the stamina usage?

Yes for most, in the case of jumping though I believe it just increases jump height
Some other exceptions are Blocking, Bows, and Woodcutting.
Most skills reduce stamina usage by 33% at max level when doing the thing, some reduce it further as well as making the thing stronger/better.