Beginner's Boards is clearly the home for this thread.
When we talk about average here, it's always after eliminating the best and worst solves. Typically they are an average of 5 (3 of the 5 solves, eliminate best and worst and average the rest [xx.xx time + xx.xx time + xx.xx time, divide by 3]), and an average of 12 (10 of the 12, etc.)
BEFORE ELIMINATING the best and the worst solves, whatever is there does not matter in any way.
AFTER ELIMINATING the best and worst solves, a DNF being there will automatically make the average DNF.
Basically, if all of the remaining solves are NOT times, you're pretty much screwed.
@b0ut POPs: No matter how big, you can always choose to fix the pop. Solve the cube as far as you can, and then you can make any needed changes to the cube and put it down to finish your solve. No penalty, just a crappy time.
If you stop the timer after a pop (without fixing it), you obviously get a DNF.