How can I check the last time stats was run on Oracle without using OEM

All of the following data dictionary tables have a LAST_ANALYZED column (replace * with USER/ALL/DBA as appropriate:

*_TABLES *_TAB_PARTITIONS *_TAB_SUBPARTITIONS *_INDEXES *_IND_PARTITIONS *_IND_SUBPARTITIONS

(There‘s lots more in the histograms fields,but I‘m not going that deep.)

Conversely, ALL_TAB_MODIFICATIONS shows rows inserted/updated/deleted (or the timestamp on which a table/partition/subpartition was truncated) since it had optimizer statistics gathered.