We discuss the mixture distribution based data-driven robust chance constrained problem.We construct a data-driven mixture distribution based uncertainty set from the perspective of simultaneously estimating higher order moments. Then, we derive a reformulation of the data-driven robust chance constrained problem.As the reformulation is not a convex programming problem, we propose new and tight convex approximations based on the piecewise linear approximation method. We establish the theoretical foundation for these approximations. Finally, numerical results show that the proposed approximations are practical and efficient.