  Stochastic rounding (SR) offers an alternative to the deterministic IEEE-754
floating-point rounding modes. In some applications such as PDEs, ODEs and
neural networks, SR empirically improves the numerical behavior and convergence
to accurate solutions while no sound theoretical background has been provided.
Recent works by Ipsen, Zhou, Higham, and Mary have computed SR probabilistic
error bounds for basic linear algebra kernels. For example, the inner product
SR probabilistic bound of the forward error is proportional to $\sqrt$ nu
instead of nu for the default rounding mode. To compute the bounds, these works
show that the errors accumulated in computation form a martingale. This paper
proposes an alternative framework to characterize SR errors based on the
computation of the variance. We pinpoint common error patterns in numerical
algorithms and propose a lemma that bounds their variance. For each probability
and through Bienaym{\'e}-Chebyshev inequality, this bound leads to better
probabilistic error bound in several situations. Our method has the advantage
of providing a tight probabilistic bound for all algorithms fitting our model.
We show how the method can be applied to give SR error bounds for the inner
product and Horner polynomial evaluation.
