Results of mechanical tests, especially in dynamics, obtained under similar input conditions always have a statistical scatter. Currently, there is no generally accepted engineering approach to the correct assessment of critical stresses for a material under dynamic loads. For example, there is no clear understanding of how many specimens need to be tested to estimate a specific ultimate stress for a specific impact rate. The present study enhances the applicability of the randomized algorithm of Sign-Perturbed Sums to solve the mentioned problem of the dynamic strength evaluation under typical restrictions on experimental results such as few numbers of data points with unknown random noises. The combination of the incubation time criteria and this new probabilistic approach permits to estimate accurately material strength parameters: the critical failure stress and the incubation time with a given level of confidence. Comparative analysis of developing approach and different modifications of the scaling law obtained by Kimberly and Ramesh showed that both models are well agreed with empirically observed data. Experimental data of dynamic test on split Hopkinson pressure bars and a drop tower setup for different quasi-brittle materials are taken to demonstrate advantages of the new method. It is shown that incubation time of failure and critical stress values of certain materials can be estimated with a high level of confidence. These results can be used to optimize experimental studies and develop new standards for dynamic testing of materials.