diff --git a/datafusion/core/tests/sqllogictests/test_files/aggregate.slt b/datafusion/core/tests/sqllogictests/test_files/aggregate.slt
index 7a1b012b84108..611fd75eff9ee 100644
--- a/datafusion/core/tests/sqllogictests/test_files/aggregate.slt
+++ b/datafusion/core/tests/sqllogictests/test_files/aggregate.slt
@@ -1091,3 +1091,15 @@ query U
 SELECT ARRAY_AGG([1]);
 ----
 [[1]]
+
+# variance_single_value
+query RRRR
+select var(sq.column1), var_pop(sq.column1), stddev(sq.column1), stddev_pop(sq.column1) from (values (1.0)) as sq;
+----
+NULL 0 NULL 0
+
+# variance_two_values
+query RRRR
+select var(sq.column1), var_pop(sq.column1), stddev(sq.column1), stddev_pop(sq.column1) from (values (1.0), (3.0)) as sq;
+----
+2 1 1.4142135623730951 1
diff --git a/datafusion/physical-expr/src/aggregate/stddev.rs b/datafusion/physical-expr/src/aggregate/stddev.rs
index 05dc56cff2c6f..4c9e46644a746 100644
--- a/datafusion/physical-expr/src/aggregate/stddev.rs
+++ b/datafusion/physical-expr/src/aggregate/stddev.rs
@@ -307,8 +307,8 @@ mod tests {
             "bla".to_string(),
             DataType::Float64,
         ));
-        let actual = aggregate(&batch, agg);
-        assert!(actual.is_err());
+        let actual = aggregate(&batch, agg).unwrap();
+        assert_eq!(actual, ScalarValue::Float64(None));
 
         Ok(())
     }
@@ -341,9 +341,8 @@ mod tests {
             "bla".to_string(),
             DataType::Float64,
         ));
-        let actual = aggregate(&batch, agg);
-        assert!(actual.is_err());
-
+        let actual = aggregate(&batch, agg).unwrap();
+        assert_eq!(actual, ScalarValue::Float64(None));
         Ok(())
     }
 
diff --git a/datafusion/physical-expr/src/aggregate/variance.rs b/datafusion/physical-expr/src/aggregate/variance.rs
index d1ccea7e1d7a5..2895137447114 100644
--- a/datafusion/physical-expr/src/aggregate/variance.rs
+++ b/datafusion/physical-expr/src/aggregate/variance.rs
@@ -286,17 +286,17 @@ impl Accumulator for VarianceAccumulator {
             }
         };
 
-        if count <= 1 {
-            return Err(DataFusionError::Internal(
-                "At least two values are needed to calculate variance".to_string(),
-            ));
-        }
-
-        if self.count == 0 {
-            Ok(ScalarValue::Float64(None))
-        } else {
-            Ok(ScalarValue::Float64(Some(self.m2 / count as f64)))
-        }
+        Ok(ScalarValue::Float64(match self.count {
+            0 => None,
+            1 => {
+                if let StatsType::Population = self.stats_type {
+                    Some(0.0)
+                } else {
+                    None
+                }
+            }
+            _ => Some(self.m2 / count as f64),
+        }))
     }
 
     fn size(&self) -> usize {
@@ -382,8 +382,8 @@ mod tests {
             "bla".to_string(),
             DataType::Float64,
         ));
-        let actual = aggregate(&batch, agg);
-        assert!(actual.is_err());
+        let actual = aggregate(&batch, agg).unwrap();
+        assert_eq!(actual, ScalarValue::Float64(None));
 
         Ok(())
     }
@@ -416,8 +416,8 @@ mod tests {
             "bla".to_string(),
             DataType::Float64,
         ));
-        let actual = aggregate(&batch, agg);
-        assert!(actual.is_err());
+        let actual = aggregate(&batch, agg).unwrap();
+        assert_eq!(actual, ScalarValue::Float64(None));
 
         Ok(())
     }