fix(rolling.sem): correct formula for sem method

Now divides by `sqrt(n * (n - ddof))`
instead of `sqrt((n - 1) * (n - ddof))`.

Author: Alvaro-Kothe

pandas/core/window/rolling.py

@@ -1779,8 +1779,8 @@ def skew(self, numeric_only: bool = False):
     def sem(self, ddof: int = 1, numeric_only: bool = False):
         # Raise here so error message says sem instead of std
         self._validate_numeric_only("sem", numeric_only)
-        return self.std(numeric_only=numeric_only) / (
-            self.count(numeric_only=numeric_only) - ddof
+        return self.std(numeric_only=numeric_only, ddof=ddof) / (
+            self.count(numeric_only=numeric_only)
         ).pow(0.5)
 
     def kurt(self, numeric_only: bool = False):
@@ -2941,16 +2941,16 @@ def sem(self, ddof: int = 1, numeric_only: bool = False):
         --------
         >>> s = pd.Series([0, 1, 2, 3])
         >>> s.rolling(2, min_periods=1).sem()
-        0         NaN
-        1    0.707107
-        2    0.707107
-        3    0.707107
+        0    NaN
+        1    0.5
+        2    0.5
+        3    0.5
         dtype: float64
         """
         # Raise here so error message says sem instead of std
         self._validate_numeric_only("sem", numeric_only)
-        return self.std(numeric_only=numeric_only) / (
-            self.count(numeric_only) - ddof
+        return self.std(numeric_only=numeric_only, ddof=ddof) / (
+            self.count(numeric_only)
         ).pow(0.5)
 
     def kurt(self, numeric_only: bool = False):

pandas/tests/window/test_groupby.py

@@ -673,17 +673,24 @@ def test_groupby_rolling_count_closed_on(self, unit):
         tm.assert_series_equal(result, expected)
 
     @pytest.mark.parametrize(
-        ("func", "kwargs"),
-        [("rolling", {"window": 2, "min_periods": 1}), ("expanding", {})],
+        ("func", "kwargs", "expected_values"),
+        [
+            (
+                "rolling",
+                {"window": 2, "min_periods": 1},
+                [np.nan, 0.5, np.nan, 0.5, 0.5],
+            ),
+            ("expanding", {}, [np.nan, 0.5, np.nan, 0.5, (1 / 3) ** 0.5]),
+        ],
     )
-    def test_groupby_rolling_sem(self, func, kwargs):
+    def test_groupby_rolling_sem(self, func, kwargs, expected_values):
         # GH: 26476
         df = DataFrame(
             [["a", 1], ["a", 2], ["b", 1], ["b", 2], ["b", 3]], columns=["a", "b"]
         )
         result = getattr(df.groupby("a"), func)(**kwargs).sem()
         expected = DataFrame(
-            {"a": [np.nan] * 5, "b": [np.nan, 0.70711, np.nan, 0.70711, 0.70711]},
+            {"a": [np.nan] * 5, "b": expected_values},
             index=MultiIndex.from_tuples(
                 [("a", 0), ("a", 1), ("b", 2), ("b", 3), ("b", 4)], names=["a", None]
             ),

pandas/tests/window/test_rolling.py

@@ -225,15 +225,10 @@ def test_datetimelike_centered_selections(
         index=date_range("2020", periods=5),
     )
 
-    if func_name == "sem":
-        kwargs = {"ddof": 0}
-    else:
-        kwargs = {}
-
     result = getattr(
         df_time.rolling("2D", closed=closed, min_periods=1, center=True),
         func_name,
-    )(**kwargs)
+    )()
 
     tm.assert_frame_equal(result, expected, check_dtype=False)
 
@@ -1078,7 +1073,7 @@ def test_rolling_sem(frame_or_series):
     result = obj.rolling(2, min_periods=1).sem()
     if isinstance(result, DataFrame):
         result = Series(result[0].values)
-    expected = Series([np.nan] + [0.7071067811865476] * 2)
+    expected = Series([np.nan] + [0.5] * 2)
     tm.assert_series_equal(result, expected)