Previously quadrature approximations were developed to determine the moments of a distribution of the response of a multivariable function when each of the variables is a random variable from a normal distribution. The error was shown to be of the order of the sixth power of the standard deviations of the random variables, but a more useful bound is desired in applied work. Only limited success has been achieved in this direction. It is shown that the best approximating distribution is a Beta distribution of the first kind with β2equal to 3 and mean, variance, and β1obtained by quadrature, where β1and β2are standard measures of skewness and kurtosis, respectively. A parametric study of the functionX=Co(ao, ±y1±y2± … ±yn)m+bowhere theyiall have the same standard deviation, σ, is conducted both analytically and by quadrature. The mean and variance obtained by quadrature are essentially exact in the range of interest. It is shown that for a large range of σ the above distribution is both a good approximation and a much better approximation than either a normal approximation with the same mean and variance or a linear approximation. The example also shows that the β2obtained by quadrature is a poor indicator of the precision of the quadrature approximation.