Why would a girl look down when I walk past her? When I am walking toward her, she will look at the ground. She doesn't smile when she looks down. She doesn't usually walk with her head down, and she's very outgoing.

This girl probably likes you or finds you attractive, but is shy about liking you or is afraid that you wouldn't like her back. Her looking down and not smiling could just be a defense mechanism to hide her feelings and protect them from potential rejection. I know you said that she's outgoing but even some of the most outgoing girls behave this way sometimes, especially towards someone they really like. Trust me on this because I'm one of those girls. I'm pretty outgoing with everyone but when confronted with a guy I like who isn't part of my circle of friends, I tend to chicken out from his gaze and look away without smiling. It's honestly a defense mechanism so then he doesn't think I'm a creep or something for randomly liking him when we aren't friends.

So yeah, she most likely likes you, but don't automatically assume so just yet. If you really want to know, try talking to her. She may seem like she wants to avoid you but if she truly likes you, she'd be willing to talk to you as long as you don't make things awkward for her. Try joking with her, flirting with her, and see if she loosens up to her outgoing self. But if she's really unwilling to talk to you and doesn't seem to loose up, then she either has extreme anxiety over guys she likes or she's not that into you. Either way, it's probably best to just leave her be because you're not going to get anywhere with a girl like that.

Anyway good luck!