Qualitative and quantitative stability with respect to probability distributions known up to their location parameter is investigated both for nonlinear two stage stochastic programs and for stochastic programs with joint probability constraints and random right hand sides. Under assumptions that guarantee convexity of the relevant nonlinear programs, Lipschitzian property of their objective functions, compactness of the sets of feasible solutions and uniqueness of the optimal solutions, the author proves consistence of the optimal values and of the optimal solutions of the corresponding approximate stochastic programs based on consistently estimated location parameters and obtains the rate of this convergence.