In this paper, we present the Gauss-type quadrature formula as a rigorous method for statistical moment estimation involving arbitrary input distributions and further extend its use to robust design optimization. The mathematical background of the Gauss-type quadrature formula is introduced and its relation with other methods such as design of experiments (DOE) and point estimate method (PEM) is discussed. Methods for constructing one dimensional Gauss-type quadrature formula are summarized and the insights are provided. To improve the efficiency of using it for robust design optimization, a semi-analytic design sensitivity analysis with respect to the statistical moments is proposed for two different multi-dimensional integration methods, the tensor product quadrature (TPQ) formula and the univariate dimension reduction (UDR) method. Through several examples, it is shown that the Gauss-type quadrature formula can be effectively used in robust design involving various non-normal distributions. The proposed design sensitivity analysis significantly reduces the number of function calls of robust optimization using the TPQ formulae, while using the UDR method, the savings of function calls are observed only in limited situations.