In this work we propose a new approach to the stability analysis of Random Boolean Networks (RBNs). In particular, we focus on two families of RBNs with k=2, in which only two subsets of canalizing Boolean function are allowed, and we show that the usual measure of RBNs stability – sometimes known as the Derrida parameter (DP) – is similar in the two cases, while their dynamics (e.g. number of attractors, length of cycles, number of frozen nodes) are different. For this reason we have introduced a new measure, that we have called attractor sensitivity (AS), computed in a way similar to DP, but perturbing only the attractors of the networks. It is proven that AS turns out to be different in the two cases analyzed. Finally, we investigate Boolean networks with k=3, tailored to solve the Density Classification Problem, and we show that also in this case the AS describes the system dynamical stability.