Level-headed == calm and sensible, logical and rational
A level-headed person is someone who is calm, sensible, and rational, especially in difficult or stressful situations, and someone who is a good problem-solver, a steady leader, or a reliable friend. They have sound judgment, can make good decisions, and are not easily swayed by emotions or bias.
- From online with editing
-------------
Based on the explanation above, do you (and your friends/family) think you are a level-headed person? ;-)
I would think everything is relative. I myself can be viewed as level headed by some people, but close friends and family also know my other side: very sensitive, at times emotional and a bit judgemental.
One true story: I always love natural stones for whatever reason. So when a builder was about to start removing a big rock in our front yard (on the boundary with the to-be-built neighbour lot), I ran out and stood right in front of the moving bulldozer, stopped the big machine.
My hubby watched the event unfold in terror, and went out to grab me. But I didn't budge till the builder came to me and explained why.
Life moved on: the rock was removed, the house was built. Then one day, the builder showed up at our door with a big rock on his machine, and asked me where to put it. Being much more level-headed then, I thanked him and was very happy to have a new rock even bigger than the one before (it's raining & dark now, otherwise I would take a pic of the rock to show you ;-)
But, please believe me, I will be very level-headed when and if a big earthquake happens!!! 
