In this paper it is analized how the symmetrical nonlinearity of restoring force and the mean square value of a random loading have influence upon the mean square value and spectrum of displacement in a vibration system which has a single degree of freedom. In this analysis an iteration method is applied to solve the equation of motion, assuming the nonlinearity and loading to have small values, and the first approximate mean square value and spectrum of displacement are derived. In case of hard spring, the mean square value of deflection is smaller than that of linear spring, while in case of soft spring it is larger than that of linear spring. These results are remarkable at the system under heavy loading. The spectrum of displacement has a peak value at the higher frequency in case of hard spring and at the lower frequency in case of soft spring.