diff --git a/src/backend/access/heap/vacuumlazy.c b/src/backend/access/heap/vacuumlazy.c
index 4b65205cd13..a65dcbebfa8 100644
--- a/src/backend/access/heap/vacuumlazy.c
+++ b/src/backend/access/heap/vacuumlazy.c
@@ -1738,7 +1738,8 @@ lazy_scan_heap(Relation onerel, VacuumParams *params, LVRelStats *vacrelstats,
 		end_parallel_vacuum(indstats, lps, nindexes);
 
 	/* Update index statistics */
-	update_index_statistics(Irel, indstats, nindexes);
+	if (vacrelstats->useindex)
+		update_index_statistics(Irel, indstats, nindexes);
 
 	/* If no indexes, make log report that lazy_vacuum_heap would've made */
 	if (vacuumed_pages)
diff --git a/src/backend/commands/analyze.c b/src/backend/commands/analyze.c
index 7295cf02156..3a9f358dd4f 100644
--- a/src/backend/commands/analyze.c
+++ b/src/backend/commands/analyze.c
@@ -600,8 +600,16 @@ do_analyze_rel(Relation onerel, VacuumParams *params,
 								 PROGRESS_ANALYZE_PHASE_FINALIZE_ANALYZE);
 
 	/*
-	 * Update pages/tuples stats in pg_class ... but not if we're doing
-	 * inherited stats.
+	 * Update pages/tuples stats in pg_class, and report ANALYZE to the stats
+	 * collector ... but not if we're doing inherited stats.
+	 *
+	 * We assume that VACUUM hasn't set pg_class.reltuples already, even
+	 * during a VACUUM ANALYZE.  Although VACUUM often updates pg_class,
+	 * exceptions exists.  A "VACUUM (ANALYZE, INDEX_CLEANUP OFF)" command
+	 * will never update pg_class entries for index relations.  It's also
+	 * possible that an individual index's pg_class entry won't be updated
+	 * during VACUUM if the index AM returns NULL from its amvacuumcleanup()
+	 * routine.
 	 */
 	if (!inh)
 	{
@@ -609,6 +617,7 @@ do_analyze_rel(Relation onerel, VacuumParams *params,
 
 		visibilitymap_count(onerel, &relallvisible, NULL);
 
+		/* Update pg_class for table relation */
 		vac_update_relstats(onerel,
 							relpages,
 							totalrows,
@@ -617,15 +626,8 @@ do_analyze_rel(Relation onerel, VacuumParams *params,
 							InvalidTransactionId,
 							InvalidMultiXactId,
 							in_outer_xact);
-	}
 
-	/*
-	 * Same for indexes. Vacuum always scans all indexes, so if we're part of
-	 * VACUUM ANALYZE, don't overwrite the accurate count already inserted by
-	 * VACUUM.
-	 */
-	if (!inh && !(params->options & VACOPT_VACUUM))
-	{
+		/* Same for indexes */
 		for (ind = 0; ind < nindexes; ind++)
 		{
 			AnlIndexData *thisdata = &indexdata[ind];
@@ -641,20 +643,28 @@ do_analyze_rel(Relation onerel, VacuumParams *params,
 								InvalidMultiXactId,
 								in_outer_xact);
 		}
+
+		/*
+		 * Now report ANALYZE to the stats collector.
+		 *
+		 * We deliberately don't report to the stats collector when doing
+		 * inherited stats, because the stats collector only tracks per-table
+		 * stats.
+		 *
+		 * Reset the changes_since_analyze counter only if we analyzed all
+		 * columns; otherwise, there is still work for auto-analyze to do.
+		 */
+		pgstat_report_analyze(onerel, totalrows, totaldeadrows,
+							  (va_cols == NIL));
 	}
 
 	/*
-	 * Report ANALYZE to the stats collector, too.  However, if doing
-	 * inherited stats we shouldn't report, because the stats collector only
-	 * tracks per-table stats.  Reset the changes_since_analyze counter only
-	 * if we analyzed all columns; otherwise, there is still work for
-	 * auto-analyze to do.
+	 * If this isn't part of VACUUM ANALYZE, let index AMs do cleanup.
+	 *
+	 * Note that most index AMs perform a no-op as a matter of policy for
+	 * amvacuumcleanup() when called in ANALYZE-only mode.  The only exception
+	 * among core index AMs is GIN/ginvacuumcleanup().
 	 */
-	if (!inh)
-		pgstat_report_analyze(onerel, totalrows, totaldeadrows,
-							  (va_cols == NIL));
-
-	/* If this isn't part of VACUUM ANALYZE, let index AMs do cleanup */
 	if (!(params->options & VACOPT_VACUUM))
 	{
 		for (ind = 0; ind < nindexes; ind++)