After re-doing my basic math, it seems to be written like this
with
datas(criteria, qty) as( values ('a', 1) ,('a', 2) , ('a', 3), ('a', 4) ,('a', 5), ('b',1), ('b',3) )
select criteria, sqrt((sum(qty * qty) -2* sum(qty) * avg(qty) + avg(qty) * avg(qty) *count( * ))/count( * )) as stddev
from datas
group by criteria
so "stddev(x)" could just be pre-transformed into
"sqrt((sum(x * x) -2* sum(x) * avg(x) + avg(x) * avg(x) * count( * ))/count( * ))"