I have some of the auto generated _WA_Sys_ statistics in my database which have not been updated in a while (by comparison to other statistics in the same table)
The rule of thumb appears to be in tables >500 rows that statistics are updated at a 20% + 500 row change of data.
However, I can see using the following query
FROM sys.sysindexes i
JOIN sys.tables t
ON i.id = t.object_id
JOIN sys.stats s
ON s.object_id = t.object_id AND i.name = s.name
CROSS APPLY sys.dm_db_stats_properties(s.object_id,s.stats_id) p
WHERE rowmodctr > 0
ORDER BY i.rowmodctr DESC
That there is a table with a number of _WA_Sys_ statistics that are way out of date (and have a rowmodctr that is higher than 20% + 500)
If a run a query against the table and add one of the columns associated with the out of date _WA_Sys_ statistics in the WHERE clause and check the updated date of the statistic, I can see it has updated.
If i run the same query with the WHERE clause again, the statistic doesn't update
It seems like the _WA_Sys statistics update when a query is run that will use them and they are out of date?