diff --git a/changelog.d/20390.bugfix b/changelog.d/20390.bugfix
new file mode 100644
index 0000000000..23e845a77f
--- /dev/null
+++ b/changelog.d/20390.bugfix
@@ -0,0 +1 @@
+Fix the `/_synapse/admin/v1/statistics/database/rooms` Admin API failing with a 500 error when Postgres has no statistics, or no most common `room_id` values, for one of the tables it checks.
diff --git a/synapse/storage/controllers/stats.py b/synapse/storage/controllers/stats.py
index 5378797f04..0cfad96e6a 100644
--- a/synapse/storage/controllers/stats.py
+++ b/synapse/storage/controllers/stats.py
@@ -92,9 +92,15 @@ class StatsController:
 
                 txn.execute(column_sql, (table,))
                 row = txn.fetchone()
-                assert row is not None
-                vals, freqs = row
 
+                # There are no stats for a table that is empty or hasn't been
+                # analyzed yet, and Postgres leaves `most_common_vals` NULL when
+                # no value is more common than the others, e.g. if every
+                # `room_id` in the table is NULL.
+                if row is None or row[0] is None:
+                    continue
+
+                vals, freqs = row
                 for room_id, freq in zip(vals, freqs):
                     room_estimates[room_id] += int(freq * table_size)
 
diff --git a/tests/rest/admin/test_statistics.py b/tests/rest/admin/test_statistics.py
index 3dc7e5dc97..4cc29d0897 100644
--- a/tests/rest/admin/test_statistics.py
+++ b/tests/rest/admin/test_statistics.py
@@ -25,13 +25,15 @@ from twisted.web.resource import Resource
 
 import synapse.rest.admin
 from synapse.api.errors import Codes
-from synapse.rest.client import login
+from synapse.rest.client import login, room
 from synapse.server import HomeServer
+from synapse.storage.database import LoggingTransaction
 from synapse.types import JsonDict
 from synapse.util.clock import Clock
 
 from tests import unittest
 from tests.test_utils import SMALL_PNG
+from tests.utils import USE_POSTGRES_FOR_TESTS
 
 
 class UserMediaStatisticsTestCase(unittest.HomeserverTestCase):
@@ -521,3 +523,56 @@ class UserMediaStatisticsTestCase(unittest.HomeserverTestCase):
         returned_order = [row["user_id"] for row in channel.json_body["users"]]
         self.assertListEqual(expected_user_list, returned_order)
         self._check_fields(channel.json_body["users"])
+
+
+class LargestRoomsStatisticsTestCase(unittest.HomeserverTestCase):
+    if not USE_POSTGRES_FOR_TESTS:
+        skip = "Requires Postgres"
+
+    servlets = [
+        synapse.rest.admin.register_servlets,
+        login.register_servlets,
+        room.register_servlets,
+    ]
+
+    def prepare(self, reactor: MemoryReactor, clock: Clock, hs: HomeServer) -> None:
+        self.admin_user = self.register_user("admin", "pass", admin=True)
+        self.admin_user_tok = self.login("admin", "pass")
+
+        self.url = "/_synapse/admin/v1/statistics/database/rooms"
+
+    def test_tables_without_common_room_ids(self) -> None:
+        """
+        Tables without stats in `pg_stats`, or whose `room_id` column is entirely
+        NULL, have no most common values and must not break the estimate.
+        """
+        room_id = self.helper.create_room_as(self.admin_user, tok=self.admin_user_tok)
+
+        # Nothing has been analyzed yet, so there is nothing to estimate from.
+        channel = self.make_request(
+            "GET",
+            self.url,
+            access_token=self.admin_user_tok,
+        )
+        self.assertEqual(200, channel.code, msg=channel.json_body)
+        self.assertNotIn(room_id, [r["room_id"] for r in channel.json_body["rooms"]])
+
+        for i in range(5):
+            self.helper.send(room_id, f"message {i}", tok=self.admin_user_tok)
+
+        # `event_push_actions` is still empty and `event_edges.room_id` is never
+        # written, so neither gets a list of most common values.
+        def analyze_txn(txn: LoggingTransaction) -> None:
+            txn.execute("ANALYZE")
+
+        self.get_success(
+            self.hs.get_datastores().main.db_pool.runInteraction("analyze", analyze_txn)
+        )
+
+        channel = self.make_request(
+            "GET",
+            self.url,
+            access_token=self.admin_user_tok,
+        )
+        self.assertEqual(200, channel.code, msg=channel.json_body)
+        self.assertIn(room_id, [r["room_id"] for r in channel.json_body["rooms"]])