diff --git a/botorch/models/transforms/outcome.py b/botorch/models/transforms/outcome.py
index 404b09cb99..063aa04ca7 100644
--- a/botorch/models/transforms/outcome.py
+++ b/botorch/models/transforms/outcome.py
@@ -223,9 +223,9 @@ def __init__(
                 standardization (if lower, only de-mean the data).
         """
         super().__init__()
-        self.register_buffer("means", torch.zeros(*batch_shape, 1, m))
-        self.register_buffer("stdvs", torch.zeros(*batch_shape, 1, m))
-        self.register_buffer("_stdvs_sq", torch.zeros(*batch_shape, 1, m))
+        self.register_buffer("means", None)
+        self.register_buffer("stdvs", None)
+        self.register_buffer("_stdvs_sq", None)
         self._outputs = normalize_indices(outputs, d=m)
         self._m = m
         self._batch_shape = batch_shape
@@ -296,9 +296,10 @@ def subset_output(self, idcs: List[int]) -> OutcomeTransform:
             batch_shape=self._batch_shape,
             min_stdv=self._min_stdv,
         )
-        new_tf.means = self.means[..., nlzd_idcs]
-        new_tf.stdvs = self.stdvs[..., nlzd_idcs]
-        new_tf._stdvs_sq = self._stdvs_sq[..., nlzd_idcs]
+        if self.means is not None:
+            new_tf.means = self.means[..., nlzd_idcs]
+            new_tf.stdvs = self.stdvs[..., nlzd_idcs]
+            new_tf._stdvs_sq = self._stdvs_sq[..., nlzd_idcs]
         if not self.training:
             new_tf.eval()
         return new_tf
@@ -319,6 +320,13 @@ def untransform(
             - The un-standardized outcome observations.
             - The un-standardized observation noise (if applicable).
         """
+        if self.means is None:
+            raise RuntimeError(
+                "`Standardize` transforms must be called on outcome data "
+                "(e.g. `transform(Y)`) before calling `untransform`, since "
+                "means and standard deviations need to be computed."
+            )
+
         Y_utf = self.means + self.stdvs * Y
         Yvar_utf = self._stdvs_sq * Yvar if Yvar is not None else None
         return Y_utf, Yvar_utf
@@ -338,13 +346,20 @@ def untransform_posterior(self, posterior: Posterior) -> Posterior:
                 "Standardize does not yet support output selection for "
                 "untransform_posterior"
             )
+        if self.means is None:
+            raise RuntimeError(
+                "`Standardize` transforms must be called on outcome data "
+                "(e.g. `transform(Y)`) before calling `untransform_posterior`, since "
+                "means and standard deviations need to be computed."
+            )
         is_mtgp_posterior = False
         if type(posterior) is GPyTorchPosterior:
             is_mtgp_posterior = posterior._is_mt
         if not self._m == posterior._extended_shape()[-1] and not is_mtgp_posterior:
             raise RuntimeError(
-                "Incompatible output dimensions encountered for transform "
-                f"{self._m} and posterior {posterior._extended_shape()[-1]}."
+                "Incompatible output dimensions encountered. Transform has output "
+                f"dimension {self._m} and posterior has "
+                f"{posterior._extended_shape()[-1]}."
             )
 
         if type(posterior) is not GPyTorchPosterior:
diff --git a/test/models/transforms/test_outcome.py b/test/models/transforms/test_outcome.py
index 1818c55837..6cf843483b 100644
--- a/test/models/transforms/test_outcome.py
+++ b/test/models/transforms/test_outcome.py
@@ -63,6 +63,36 @@ def test_abstract_base_outcome_transform(self):
         with self.assertRaises(NotImplementedError):
             oct.untransform_posterior(None)
 
+    def test_standardize_raises_when_mean_not_set(self) -> None:
+        posterior = _get_test_posterior(
+            shape=torch.Size([1, 1]), device=self.device, dtype=torch.float64
+        )
+        for transform in [
+            Standardize(m=1),
+            ChainedOutcomeTransform(
+                chained=ChainedOutcomeTransform(stand=Standardize(m=1))
+            ),
+        ]:
+            with self.assertRaises(
+                RuntimeError,
+                msg="`Standardize` transforms must be called on outcome data "
+                "(e.g. `transform(Y)`) before calling `untransform_posterior`, since "
+                "means and standard deviations need to be computed.",
+            ):
+                transform.untransform_posterior(posterior)
+
+            new_tf = transform.subset_output([0])
+            assert isinstance(new_tf, type(transform))
+
+            y = torch.arange(3, device=self.device, dtype=torch.float64)
+            with self.assertRaises(
+                RuntimeError,
+                msg="`Standardize` transforms must be called on outcome data "
+                "(e.g. `transform(Y)`) before calling `untransform`, since "
+                "means and standard deviations need to be computed.",
+            ):
+                transform.untransform(y)
+
     def test_standardize(self):
         # test error on incompatible dim
         tf = Standardize(m=1)
@@ -208,8 +238,15 @@ def test_standardize(self):
 
             # test error on incompatible output dimension
             # TODO: add a unit test for MTGP posterior once #840 goes in
-            tf_big = Standardize(m=4).eval()
-            with self.assertRaises(RuntimeError):
+            tf_big = Standardize(m=4)
+            Y = torch.arange(4, device=self.device, dtype=dtype).reshape((1, 4))
+            tf_big(Y)
+            with self.assertRaises(
+                RuntimeError,
+                msg="Incompatible output dimensions encountered. Transform has output "
+                f"dimension {tf._m} and posterior has "
+                f"{posterior._extended_shape()[-1]}.",
+            ):
                 tf_big.untransform_posterior(posterior2)
 
         # test transforming a subset of outcomes