What would cause a child to be loud and unruly at home and completely mute at school??

The boy is a home devil and a street angel. This is a  common phase that most boys go through when they need acceptance and attention at home but put on a mask at school.

Child has no fear of rejection at home, child knows he is loved, no matter what child does. Not same at school. As child grows a little older child will do well academically, do not expect child to be mute for long at school. 

people are very different when around people not their family

He feels at home.In school he doesn't.

Plus he may have social anxiety.

Social anxiety and fear of authority. He works so hard to not look foolish in front of teachers and classmates and he saves up all that pent up anxiety for people that he knows won't judge him. Encourage him to feel confident at school by working hard on his work and set boundaries at home.