This paper presents in detail the originally developed Quadratic Point Estimate Method (QPEM), aimed at efficiently and accurately computing the first four output moments of probabilistic distributions, using 2n^2+1 sample (or sigma) points, with n, the number of input random variables. The proposed QPEM particularly offers an effective, superior, and practical alternative to existing sampling and quadrature methods for low- and moderately-high-dimensional problems. Detailed theoretical derivations are provided proving that the proposed method can achieve a fifth or higher-order accuracy for symmetric input distributions. Various numerical examples, from simple polynomial functions to nonlinear finite element analyses with random field representations, support the theoretical findings and further showcase the validity, efficiency, and applicability of the QPEM, from low- to high-dimensional problems.