The article is devoted to the study of the behavior of the quasi-random integration remainder in the calculation of high-dimensional integrals. As noted in the previous work of the authors, the asymptotic behavior of its decrease, determined by the Koksma-Hlawka inequality, can be used only with a very large number of integration nodes N, which cannot be implemented on modern computers. The article introduces the concept of a mean order of decreasing remainder, which makes it possible to judge its properties with the N values available for realization and to compare various pseudo-random sequences. A number of numerical examples are given. In all cases, it turned out that the Sobol’ sequences in the sense of this criterion are somewhat better than the Holton sequences.