Stats arguments always give way to probability arguments. But, alas! Where is a Cubs fan to get rapid-fire assessment of whether a guy is doing differently in one situation or another?!?!?
So, here is R-code for doing just that. It's not horribly user friendly, but it's not horribly user cruel, either. You can download R for free at: http://www.r-project.org. Good R programmers make this stuff really easy to use. I learned to program using punch cards while listening to music on vinyl and wondering why my polyester shirt was so uncomfortable, so screw that.
Run all that through R and you'll get this plot. The number at the very bottom tells you that one in 1.48 players would show this difference just by chance alone if he was basically the same batter all the time.
You can replace the situations with whatever you please.