[{"_id":{"state":"NY","st":"value"},"List":[{"id":"21","score":18.75,"name":"PU"},{"id":"21","score":25.0,"name":"PU"},{"id":"23","score":25.0,"name":"CL"},{"id":"23","score":56.25,"name":"CL"}]}]

db.collection.aggregate([{"$unwind":{"path":"$List"}},{"$group":{"_id":{"state":"$_id.state","st":"$_id.st","id":"$List.id","name":"$List.name"},"avg":{"$avg":"$List.score"}}},{"$project":{"_id":{"state":"$_id.state","st":"$_id.st"},"List":{"name":"$_id.name","id":"$_id.id","avg":"$avg"}}},{"$group":{"_id":"$_id","List":{"$push":"$List"}}}])