you can compute the variance of field foo by using:
select
SELECT
(sum(foo*foo)-sum(foo)*sum(foo)/count(foo))/count(foo) AS variance
FROM ...
standard deviation is just the square root of the variance (you can use the sqrt() function).
Regarding percentiles, there is no way to do this currently. The underlying query engine that BigQuery uses has a quantiles function, so it should be relatively straightforward to add. Note that this would be a statistical approximation, since exact percentiles are very expensive to compute over large data. I'll ping you back after investigating further to make sure there are no gotchas with exposing the additional function.
Moreover, given that we've had several requests, we might add shortcuts for mean/median/variance queries (if these would make your life easier, please let us know).
-jordan