diff --git a/pandas/core/methods/describe.py b/pandas/core/methods/describe.py
index 17d4d38c97f33..df38af68e3a87 100644
--- a/pandas/core/methods/describe.py
+++ b/pandas/core/methods/describe.py
@@ -226,9 +226,9 @@ def describe_numeric_1d(series: Series, percentiles: Sequence[float]) -> Series:
         The percentiles to include in the output.
     """
     from pandas import Series
-
+    
     formatted_percentiles = format_percentiles(percentiles)
-
+    
     stat_index = ["count", "mean", "std", "min"] + formatted_percentiles + ["max"]
     d = (
         [series.count(), series.mean(), series.std(), series.min()]
@@ -345,19 +345,22 @@ def _refine_percentiles(
     percentiles : list-like of numbers, optional
         The percentiles to include in the output.
     """
+
+    from pandas import Series
+
     if percentiles is None:
         return np.array([0.25, 0.5, 0.75])
 
+    # Handling empty list , empty numpy array and empty Series :
+    elif isinstance(percentiles, (list, np.ndarray, Series)) and len(percentiles) == 0:
+        return np.array([])
+
     # explicit conversion of `percentiles` to list
     percentiles = list(percentiles)
 
     # get them all to be in [0, 1]
     validate_percentile(percentiles)
 
-    # median should always be included
-    if 0.5 not in percentiles:
-        percentiles.append(0.5)
-
     percentiles = np.asarray(percentiles)
 
     # sort and check for duplicates
diff --git a/pandas/io/formats/format.py b/pandas/io/formats/format.py
index b7fbc4e5e22b7..13eab73e197b6 100644
--- a/pandas/io/formats/format.py
+++ b/pandas/io/formats/format.py
@@ -1565,6 +1565,7 @@ def format_percentiles(
     >>> format_percentiles([0, 0.5, 0.02001, 0.5, 0.666666, 0.9999])
     ['0%', '50%', '2.0%', '50%', '66.67%', '99.99%']
     """
+
     percentiles = np.asarray(percentiles)
 
     # It checks for np.nan as well
@@ -1575,7 +1576,13 @@ def format_percentiles(
     ):
         raise ValueError("percentiles should all be in the interval [0,1]")
 
-    percentiles = 100 * percentiles
+    # Fix for issue #60550
+    if len(percentiles) > 0 :
+       percentiles = 100 * percentiles 
+
+    else : 
+       percentiles = np.array([])
+    
     prec = get_precision(percentiles)
     percentiles_round_type = percentiles.round(prec).astype(int)
 
@@ -1595,6 +1602,10 @@ def format_percentiles(
 
 
 def get_precision(array: np.ndarray | Sequence[float]) -> int:
+    # Fix for issue #60550
+    if array.size == 0:
+       return 0
+
     to_begin = array[0] if array[0] > 0 else None
     to_end = 100 - array[-1] if array[-1] < 100 else None
     diff = np.ediff1d(array, to_begin=to_begin, to_end=to_end)
diff --git a/pandas/tests/frame/methods/test_describe.py b/pandas/tests/frame/methods/test_describe.py
index e9206e86b7b08..79736424067bb 100644
--- a/pandas/tests/frame/methods/test_describe.py
+++ b/pandas/tests/frame/methods/test_describe.py
@@ -413,3 +413,59 @@ def test_describe_exclude_pa_dtype(self):
             dtype=pd.ArrowDtype(pa.float64()),
         )
         tm.assert_frame_equal(result, expected)
+
+    def test_describe_empty_percentiles(self):
+        # 60550 :
+        # Create a simple DataFrame
+        df = DataFrame({"a": [1, 2, 3, 4, 5]})
+
+        # Case 1: Passing an empty list
+        result = df.describe(percentiles=[])
+        expected = DataFrame(
+            {"a": [5, 3, 1.581139, 1, 5]},
+            index=["count", "mean", "std", "min", "max"],
+        )
+        tm.assert_frame_equal(result, expected)
+
+        # Case 2: Passing an empty numpy array
+        result = df.describe(percentiles=np.array([]))
+        tm.assert_frame_equal(result, expected)
+
+    def test_describe_with_single_percentile(self):
+        # 60550 :
+        # Create a simple DataFrame
+        df = DataFrame({"a": [1, 2, 3, 4, 5]})
+        # Case 1: Passing a single percentile
+        result = df.describe(percentiles=[0.5])
+        expected = DataFrame(
+            {"a": [5, 3, 1.581139, 1, 3, 5]},
+            index=["count", "mean", "std", "min", "50%", "max"],
+        )
+        tm.assert_frame_equal(result, expected)
+
+    def test_describe_empty_numpy_percentile(self):
+        df = DataFrame({"a": [1, 2, 3, 4, 5]})
+
+        # Passing empty NumPy array as percentiles
+        result = df.describe(percentiles=np.array([]))
+
+        # Expected output should not include percentiles
+        expected = DataFrame(
+            {"a": [5, 3.0, 1.581139, 1, 5]},
+            index=["count", "mean", "std", "min", "max"],
+        )
+        tm.assert_frame_equal(result, expected)
+
+    def test_describe_empty_series_percentile(self):
+        df = DataFrame({"a": [1, 2, 3, 4, 5]})
+
+        # Passing empty Series as percentiles
+        result = df.describe(percentiles=Series([], dtype=float))
+
+        # Expected output should not include percentiles
+        expected = DataFrame(
+            {"a": [5, 3.0, 1.581139, 1, 5]},
+            index=["count", "mean", "std", "min", "max"],
+        )
+
+        tm.assert_frame_equal(result, expected)