diff --git a/news/8654.bugfix b/news/8654.bugfix
new file mode 100644
index 00000000000..ec0df7a903e
--- /dev/null
+++ b/news/8654.bugfix
@@ -0,0 +1,2 @@
+Trace a better error message on installation failure due to invalid ``.data``
+files in wheels.
diff --git a/src/pip/_internal/operations/install/wheel.py b/src/pip/_internal/operations/install/wheel.py
index 8f73a88b074..681fc0aa8ef 100644
--- a/src/pip/_internal/operations/install/wheel.py
+++ b/src/pip/_internal/operations/install/wheel.py
@@ -583,8 +583,28 @@ def data_scheme_file_maker(zip_file, scheme):
         def make_data_scheme_file(record_path):
             # type: (RecordPath) -> File
             normed_path = os.path.normpath(record_path)
-            _, scheme_key, dest_subpath = normed_path.split(os.path.sep, 2)
-            scheme_path = scheme_paths[scheme_key]
+            try:
+                _, scheme_key, dest_subpath = normed_path.split(os.path.sep, 2)
+            except ValueError:
+                message = (
+                    "Unexpected file in {}: {!r}. .data directory contents"
+                    " should be named like: '<scheme key>/<path>'."
+                ).format(wheel_path, record_path)
+                raise InstallationError(message)
+
+            try:
+                scheme_path = scheme_paths[scheme_key]
+            except KeyError:
+                valid_scheme_keys = ", ".join(sorted(scheme_paths))
+                message = (
+                    "Unknown scheme key used in {}: {} (for file {!r}). .data"
+                    " directory contents should be in subdirectories named"
+                    " with a valid scheme key ({})"
+                ).format(
+                    wheel_path, scheme_key, record_path, valid_scheme_keys
+                )
+                raise InstallationError(message)
+
             dest_path = os.path.join(scheme_path, dest_subpath)
             assert_no_path_traversal(scheme_path, dest_path)
             return ZipBackedFile(record_path, dest_path, zip_file)
diff --git a/tests/functional/test_install_wheel.py b/tests/functional/test_install_wheel.py
index c53f13ca415..ad4e749676f 100644
--- a/tests/functional/test_install_wheel.py
+++ b/tests/functional/test_install_wheel.py
@@ -681,3 +681,36 @@ def test_correct_package_name_while_creating_wheel_bug(script, package_name):
     package = create_basic_wheel_for_package(script, package_name, '1.0')
     wheel_name = os.path.basename(package)
     assert wheel_name == 'simple_package-1.0-py2.py3-none-any.whl'
+
+
+@pytest.mark.parametrize("name", ["purelib", "abc"])
+def test_wheel_with_file_in_data_dir_has_reasonable_error(
+    script, tmpdir, name
+):
+    """Normally we expect entities in the .data directory to be in a
+    subdirectory, but if they are not then we should show a reasonable error
+    message that includes the path.
+    """
+    wheel_path = make_wheel(
+        "simple", "0.1.0", extra_data_files={name: "hello world"}
+    ).save_to_dir(tmpdir)
+
+    result = script.pip(
+        "install", "--no-index", str(wheel_path), expect_error=True
+    )
+    assert "simple-0.1.0.data/{}".format(name) in result.stderr
+
+
+def test_wheel_with_unknown_subdir_in_data_dir_has_reasonable_error(
+    script, tmpdir
+):
+    wheel_path = make_wheel(
+        "simple",
+        "0.1.0",
+        extra_data_files={"unknown/hello.txt": "hello world"}
+    ).save_to_dir(tmpdir)
+
+    result = script.pip(
+        "install", "--no-index", str(wheel_path), expect_error=True
+    )
+    assert "simple-0.1.0.data/unknown/hello.txt" in result.stderr