  The main objective of this work is to investigate non-linear errors and
pairwise summation using stochastic rounding (SR) in variance computation
algorithms. We estimate the forward error of computations under SR through two
methods: the first is based on a bound of the variance and
Bienaym{\'e}-Chebyshev inequality, while the second is based on martingales and
Azuma-Hoeffding inequality. The study shows that for pairwise summation, using
SR results in a probabilistic bound of the forward error proportional to
log(n)u rather than the deterministic bound in O(log(n)u) when using the
default rounding mode. We examine two algorithms that compute the variance,
called ''textbook'' and ''two-pass'', which both exhibit non-linear errors.
Using the two methods mentioned above, we show that these algorithms' forward
errors have probabilistic bounds under SR in O($\sqrt$ nu) instead of nu for
the deterministic bounds. We show that this advantage holds using pairwise
summation for both textbook and two-pass, with probabilistic bounds of the
forward error proportional to log(n)u.
