Do people know if you're online on facebook outside of chat?

I'm new to facebook. I always keep my chat signed off but it seems like maybe people know I'm online if I'm just signed in at all. I prefer my anonymity and full privacy, so I was wondering if this is a coincidence or if there's actually a way that people know if you're simply signed in and if so, a way I can prevent certain or all people from knowing? There's nothing malicious or anything going on here, I just prefer privacy. Thanks.