Does it take a bigger person to stay and argue or removing themselves from the situation?

You're always the bigger person when walking away from an argument. By doing so, you're not being a baby and adding more drama to the situation. There are some places where you would be the bigger person, like if you were sticking up for someone that was being bullied or if you were sticking up for yourself for being bullied. Other than that, I really wouldn't see why someone should stay and argue...it just causes more argument and makes people farther and farther away from each other...and destroys relationships. I think it's good that you remove yourself from the situation; it shows maturity. :)