What are statistics, under what circumstances they go out of date, how do you update them?

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Statistics determine the selectivity of the indexes. If an indexed column has unique values
then the selectivity of that index is more, as opposed to an index with non-unique values.
Query optimizer uses these indexes in determining whether to choose an index or not
while executing a query.

Some situations under which you should update statistics:
1) If there is significant change in the key values in the index
2) If a large amount of data in an indexed column has been added, changed, or removed
(that is, if the distribution of key values has changed), or the table has been truncated using
the TRUNCATE TABLE statement and then repopulated
3) Database is upgraded from a previous version