[{"_id":"xxx","name":"some_name","data":{"first_thing":[1,2,3,4],"second_thing":[5,3,2,4],}}]

db.collection.aggregate([{"$project":{"_id":"$name","mean":{"$avg":"$data.first_thing"},"std":{"$stdDevPop":"$data.first_thing"}}},{$addFields:{"data.first_thing":[{mean:"$mean",std:"$std"}]}},{$project:{mean:0,std:0}}])