How does Scout show mental courage and psychological strength?

One example of Scout's mental strength can be found in Chapter 9, when she demonstrates considerable self-discipline to pull out of a fight with Cecil Jacobs. Cecil is insulting Atticus and calling him a "disgrace," and usually this would be more than enough provocation for Scout to start fighting with him. But Scout remembers a promise she made to Atticus not to fight and accordingly walks away from the fight. When she walks away she hears "Scout's a cow-ward! ringin in (her) ears," but she still manages to maintain her self-discipline, reasoning that "If I fought Cecil I would be letting Atticus down." Given that Scout's temperament is usually quite emotional and spirited and given how much she must hate hearing her father called a "disgrace," the self-discipline she demonstrates here is really very impressive. As she herself says, "It was the first time I ever walked away from a fight."
Another example of Scout's courage, physical and mental, can be found in Chapter 15. Atticus is at the jailhouse, guarding Tom Robinson against the angry lynch mob that has assembled outside. The mob is menacing, and threatening, and one of the men grabs Jem "roughly by the collar" to move him out of his way. Scout, seeing her brother treated like this, kicks the man above the shin, sending him reeling back "in real pain." She declares, defiantly, "Ain't nobody gonna do Jem that way." Scout's loyalty to and love for her brother gives her the courage in this moment to stand up to a rather sinister bully.