diff --git a/sql/catalyst/src/main/scala/org/apache/spark/sql/catalyst/analysis/Analyzer.scala b/sql/catalyst/src/main/scala/org/apache/spark/sql/catalyst/analysis/Analyzer.scala
index 14a2a323c8852..f39ea03500636 100644
--- a/sql/catalyst/src/main/scala/org/apache/spark/sql/catalyst/analysis/Analyzer.scala
+++ b/sql/catalyst/src/main/scala/org/apache/spark/sql/catalyst/analysis/Analyzer.scala
@@ -730,6 +730,11 @@ class Analyzer(
         }
         Sort(newOrders, global, child)
 
+      // Eliminate the useless position numbers
+      case s @ Sort(orders, global, child)
+          if !conf.orderByOrdinal && orders.exists(o => IntegerIndex.unapply(o.child).nonEmpty) =>
+        Sort(orders.filterNot(o => IntegerIndex.unapply(o.child).nonEmpty), global, child)
+
       // Replace the index with the corresponding expression in aggregateExpressions. The index is
       // a 1-base position of aggregateExpressions, which is output columns (select expression)
       case a @ Aggregate(groups, aggs, child)
@@ -1252,7 +1257,9 @@ class Analyzer(
           case ae: AnalysisException => filter
         }
 
-      case sort @ Sort(sortOrder, global, aggregate: Aggregate) if aggregate.resolved =>
+      // If there exists ordinal sort orders, it's not resolved completely yet. See SPARK-16955.
+      case sort @ Sort(sortOrder, global, aggregate: Aggregate) if aggregate.resolved &&
+          sortOrder.forall(x => IntegerIndex.unapply(x.child).isEmpty) =>
 
         // Try resolving the ordering as though it is in the aggregate clause.
         try {
diff --git a/sql/core/src/test/scala/org/apache/spark/sql/SQLQuerySuite.scala b/sql/core/src/test/scala/org/apache/spark/sql/SQLQuerySuite.scala
index eac588fff2fc7..93b3fb1115901 100644
--- a/sql/core/src/test/scala/org/apache/spark/sql/SQLQuerySuite.scala
+++ b/sql/core/src/test/scala/org/apache/spark/sql/SQLQuerySuite.scala
@@ -495,6 +495,19 @@ class SQLQuerySuite extends QueryTest with SharedSQLContext {
     )
   }
 
+  test("SPARK-16955: Using ordinals in ORDER BY and GROUP BY causes an analysis error") {
+    withSQLConf(SQLConf.ORDER_BY_ORDINAL.key -> "true") {
+      checkAnswer(
+        sql("SELECT a, avg(a) FROM (SELECT * FROM VALUES 1,2,3 T(a)) U GROUP BY 1 ORDER BY 1 DESC"),
+        sql("SELECT a, avg(a) FROM (SELECT * FROM VALUES 1,2,3 T(a)) U GROUP BY a ORDER BY a DESC"))
+    }
+    withSQLConf(SQLConf.ORDER_BY_ORDINAL.key -> "false") {
+      checkAnswer(
+        sql("SELECT a, avg(a) FROM (SELECT * FROM VALUES 1,2,3 T(a)) U GROUP BY 1 ORDER BY 1 DESC"),
+        sql("SELECT a, avg(a) FROM (SELECT * FROM VALUES 1,2,3 T(a)) U GROUP BY a"))
+    }
+  }
+
   test("select *") {
     checkAnswer(
       sql("SELECT * FROM testData"),