Can I not tell my family I moved out?
That depends. What's the reasoning for not telling them you moved out?
Because I'm a very independent person and I don't care about what they think.
I understand being independent makes you feel like you only need to care about yourself. Not giving your family notice when you plan to move out is really rude and inconsiderate, you should have a chat with your family and let them know your plans for the future. Not doing this could result in trust issues with your family.
I don't care what they think about me. I'm going to move out anyways.
It is up to you but you could be causing more pain and trouble if you move out without notice. You might want to talk to them later in life but they might not want you around after doing this.