diff --git a/python/paddle/fluid/tests/unittests/test_zero_dim_tensor.py b/python/paddle/fluid/tests/unittests/test_zero_dim_tensor.py
index 94740925ea21d..64881c08fb637 100644
--- a/python/paddle/fluid/tests/unittests/test_zero_dim_tensor.py
+++ b/python/paddle/fluid/tests/unittests/test_zero_dim_tensor.py
@@ -622,6 +622,45 @@ def test_median(self):
         self.assertEqual(x.grad.shape, [])
         np.testing.assert_allclose(x.grad, 3.0)
 
+    def test_std(self):
+        x = paddle.rand([])
+        x.stop_gradient = False
+        out1 = paddle.std(x)
+        out2 = paddle.std(x, [])
+        out1.backward()
+        out2.backward()
+
+        # checkout shape of out
+        self.assertEqual(out1.shape, [])
+        self.assertEqual(out2.shape, [])
+
+        # checkout value of out
+        self.assertEqual(out1, 0)
+        self.assertEqual(out2, 0)
+
+        # checkout backward
+        self.assertEqual(x.grad.shape, [])
+
+    def test_var(self):
+        x = paddle.rand([])
+        x.stop_gradient = False
+        out1 = paddle.var(x)
+        out2 = paddle.var(x, [])
+        out1.backward()
+        out2.backward()
+
+        # checkout shape of out
+        self.assertEqual(out1.shape, [])
+        self.assertEqual(out2.shape, [])
+
+        # checkout value of out
+        self.assertEqual(out1, 0)
+        self.assertEqual(out2, 0)
+
+        # checkout backward
+        self.assertEqual(x.grad.shape, [])
+        np.testing.assert_allclose(x.grad, 0)
+
     def test_quantile(self):
         # 1) x is 0D
         x = paddle.rand([])
@@ -1708,6 +1747,48 @@ def test_median(self):
         self.assertEqual(res[2].shape, ())
         np.testing.assert_allclose(res[2], 1.0)
 
+    @prog_scope()
+    def test_std(self):
+        x = paddle.rand([])
+        out1 = paddle.std(x)
+        out2 = paddle.std(x, [])
+        paddle.static.append_backward(out1)
+        paddle.static.append_backward(out2)
+
+        prog = paddle.static.default_main_program()
+        res = self.exe.run(
+            prog,
+            fetch_list=[
+                x,
+                out1,
+                out2,
+            ],
+        )
+        self.assertEqual(res[0].shape, ())
+        self.assertEqual(res[1].shape, ())
+        self.assertEqual(res[2].shape, ())
+
+    @prog_scope()
+    def test_var(self):
+        x = paddle.rand([])
+        out1 = paddle.var(x)
+        out2 = paddle.var(x, [])
+        paddle.static.append_backward(out1)
+        paddle.static.append_backward(out2)
+
+        prog = paddle.static.default_main_program()
+        res = self.exe.run(
+            prog,
+            fetch_list=[
+                x,
+                out1,
+                out2,
+            ],
+        )
+        self.assertEqual(res[0].shape, ())
+        self.assertEqual(res[1].shape, ())
+        self.assertEqual(res[2].shape, ())
+
     @prog_scope()
     def test_quantile(self):
         x1 = paddle.rand([])
diff --git a/python/paddle/tensor/stat.py b/python/paddle/tensor/stat.py
index f9784478393dc..35d1a4337b6df 100644
--- a/python/paddle/tensor/stat.py
+++ b/python/paddle/tensor/stat.py
@@ -148,7 +148,7 @@ def var(x, axis=None, unbiased=True, keepdim=False, name=None):
         check_variable_and_dtype(x, 'x', ['float32', 'float64'], 'var')
 
     u = mean(x, axis, True, name)
-    out = paddle.sum((x - u) ** 2, axis, keepdim=keepdim, name=name)
+    out = paddle.sum(paddle.pow((x - u), 2), axis, keepdim=keepdim, name=name)
 
     dtype = x.dtype
     n = paddle.cast(paddle.numel(x), paddle.int64) / paddle.cast(
@@ -212,7 +212,6 @@ def std(x, axis=None, unbiased=True, keepdim=False, name=None):
     """
     if not in_dygraph_mode():
         check_variable_and_dtype(x, 'x', ['float32', 'float64'], 'std')
-
     out = var(**locals())
     return paddle.sqrt(out)