diff --git a/mysql-test/r/group_min_max.result b/mysql-test/r/group_min_max.result
index 91579a7ea42..b1703c51f4e 100644
--- a/mysql-test/r/group_min_max.result
+++ b/mysql-test/r/group_min_max.result
@@ -2043,3 +2043,30 @@ c1	c2
 30	8
 30	9
 drop table t1;
+CREATE TABLE t1 (a varchar(5), b int(11), PRIMARY KEY (a,b));
+INSERT INTO t1 VALUES ('AA',1), ('AA',2), ('AA',3), ('BB',1), ('AA',4);
+OPTIMIZE TABLE t1;
+Table	Op	Msg_type	Msg_text
+test.t1	optimize	status	OK
+SELECT a FROM t1 WHERE a='AA' GROUP BY a;
+a
+AA
+SELECT a FROM t1 WHERE a='BB' GROUP BY a;
+a
+BB
+EXPLAIN SELECT a FROM t1 WHERE a='AA' GROUP BY a;
+id	select_type	table	type	possible_keys	key	key_len	ref	rows	Extra
+1	SIMPLE	t1	ref	PRIMARY	PRIMARY	7	const	3	Using where; Using index
+EXPLAIN SELECT a FROM t1 WHERE a='BB' GROUP BY a;
+id	select_type	table	type	possible_keys	key	key_len	ref	rows	Extra
+1	SIMPLE	t1	ref	PRIMARY	PRIMARY	7	const	1	Using where; Using index
+SELECT DISTINCT a FROM t1 WHERE a='BB';
+a
+BB
+SELECT DISTINCT a FROM t1 WHERE a LIKE 'B%';
+a
+BB
+SELECT a FROM t1 WHERE a LIKE 'B%' GROUP BY a;
+a
+BB
+DROP TABLE t1;
diff --git a/mysql-test/t/group_min_max.test b/mysql-test/t/group_min_max.test
index e15ef92116c..8dc55532bbf 100644
--- a/mysql-test/t/group_min_max.test
+++ b/mysql-test/t/group_min_max.test
@@ -715,3 +715,24 @@ select distinct c1, c2 from t1 order by c2;
 select c1,min(c2) as c2 from t1 group by c1 order by c2;
 select c1,c2 from t1 group by c1,c2 order by c2;
 drop table t1;
+
+#
+# Bug #16203: Analysis for possible min/max optimization erroneously
+#             returns impossible range
+#
+
+CREATE TABLE t1 (a varchar(5), b int(11), PRIMARY KEY (a,b));
+INSERT INTO t1 VALUES ('AA',1), ('AA',2), ('AA',3), ('BB',1), ('AA',4);
+OPTIMIZE TABLE t1;
+
+SELECT a FROM t1 WHERE a='AA' GROUP BY a;
+SELECT a FROM t1 WHERE a='BB' GROUP BY a;
+
+EXPLAIN SELECT a FROM t1 WHERE a='AA' GROUP BY a;
+EXPLAIN SELECT a FROM t1 WHERE a='BB' GROUP BY a;
+
+SELECT DISTINCT a FROM t1 WHERE a='BB';
+SELECT DISTINCT a FROM t1 WHERE a LIKE 'B%';
+SELECT a FROM t1 WHERE a LIKE 'B%' GROUP BY a;
+
+DROP TABLE t1;
diff --git a/sql/opt_range.cc b/sql/opt_range.cc
index 7c4f5fbe218..1b712700b18 100644
--- a/sql/opt_range.cc
+++ b/sql/opt_range.cc
@@ -7780,6 +7780,7 @@ void cost_group_min_max(TABLE* table, KEY *index_info, uint used_key_parts,
     quick_prefix_selectivity= (double) quick_prefix_records /
                               (double) table_records;
     num_groups= (uint) rint(num_groups * quick_prefix_selectivity);
+    set_if_bigger(num_groups, 1);
   }
 
   if (used_key_parts > group_key_parts)