まずソート

 { "$sort": [{"name": "targetValue", "direction": 1}] },

次にgroupして標準偏差、平均値、中央値を求める

{"$group": {
  "_id": "$service",
  "avg": {"$avg": "$targetValue"},
  "targetValues": {"$push": "$targetValue"},
  "stdDev": {"$stdDevPop": "$targetValue"},
  "stdSamp": {"$stdDevSamp": "$targetValue"}
}},

第一四分位、第三四分位も求める

{"$project": {
    "_id": 1,
    "avg": 1,
    "stdDev": 1,
    "stdSamp": 1,
    "median": {
      "$arrayElemAt": ["$targetValues", {"$toInt": {
        "$divide": [{"$size": "$targetValues"}, 2]
      }}]
    },
    "1stQuarter": {
      "$arrayElemAt": ["$targetValues", {"$toInt": {
        "$divide": [{"$size": "$targetValues"}, 4]
      }}]
    },
    "3rdQuarter": {
      "$arrayElemAt": ["$targetValues",{"$toInt": {
        "$multiply": [{"$divide": [{"$size": "$targetValues"}, 4]}, 3]
      }}]
    }
  }
}

一つにして

[
{ "$sort": [{"name": "targetValue", "direction": 1}] },
{"$group": {
  "_id": "$service",
  "avg": {"$avg": "$targetValue"},
  "targetValues": {"$push": "$targetValue"},
  "stdDev": {"$stdDevPop": "$targetValue"},
  "stdSamp": {"$stdDevSamp": "$targetValue"}
}},
{"$project": {
    "_id": 1,
    "avg": 1,
    "stdDev": 1,
    "stdSamp": 1,
    "median": {
      "$arrayElemAt": ["$targetValues", {"$toInt": {
        "$divide": [{"$size": "$targetValues"}, 2]
      }}]
    },
    "1stQuarter": {
      "$arrayElemAt": ["$targetValues", {"$toInt": {
        "$divide": [{"$size": "$targetValues"}, 4]
      }}]
    },
    "3rdQuarter": {
      "$arrayElemAt": ["$targetValues",{"$toInt": {
        "$multiply": [{"$divide": [{"$size": "$targetValues"}, 4]}, 3]
      }}]
    }
  }
}
]

注意点

mongoのバージョンはv4以上じゃないと$toIntが使えないです。