diff --git a/lib/iris/tests/unit/util/test__coord_regular.py b/lib/iris/tests/unit/util/test__coord_regular.py
index e5141419ec..a5e9aca9ed 100644
--- a/lib/iris/tests/unit/util/test__coord_regular.py
+++ b/lib/iris/tests/unit/util/test__coord_regular.py
@@ -94,6 +94,18 @@ def test_irregular_points(self):
         self.assertEqual(exp_avdiff, result_avdiff)
         self.assertFalse(result)
 
+    def test_single_point(self):
+        lone_point = np.array([4])
+        result_avdiff, result = points_step(lone_point)
+        self.assertTrue(np.isnan(result_avdiff))
+        self.assertTrue(result)
+
+    def test_no_points(self):
+        no_points = np.array([])
+        result_avdiff, result = points_step(no_points)
+        self.assertTrue(np.isnan(result_avdiff))
+        self.assertTrue(result)
+
 
 if __name__ == "__main__":
     tests.main()
diff --git a/lib/iris/util.py b/lib/iris/util.py
index 826162fa7f..bc8d27be8f 100644
--- a/lib/iris/util.py
+++ b/lib/iris/util.py
@@ -1397,11 +1397,28 @@ def regular_step(coord):
 
 
 def points_step(points):
-    """Determine whether a NumPy array has a regular step."""
-    diffs = np.diff(points)
-    avdiff = np.mean(diffs)
-    # TODO: This value for `rtol` is set for test_analysis to pass...
-    regular = np.allclose(diffs, avdiff, rtol=0.001)
+    """Determine whether `points` has a regular step.
+
+    Parameters
+    ----------
+    points : numeric, array-like
+        The sequence of values to check for a regular difference.
+
+    Returns
+    -------
+    numeric, bool
+        A tuple containing the average difference between values, and whether the difference is regular.
+    """
+    # Calculations only make sense with multiple points
+    points = np.asanyarray(points)
+    if points.size >= 2:
+        diffs = np.diff(points)
+        avdiff = np.mean(diffs)
+        # TODO: This value for `rtol` is set for test_analysis to pass...
+        regular = np.allclose(diffs, avdiff, rtol=0.001)
+    else:
+        avdiff = np.nan
+        regular = True
     return avdiff, regular