In the Equation-free framework, a macro-coarse projective integration method consists of two parts: the time stepper and time projection on macro scale. The first one consists of lifting, micro simulation and restriction. For extracting directly from microscopic simulations the information which would be obtained from the macroscopic model of two-dimensional microscopic systems, the time stepper based on the one-dimensional cumulative distribution functions, the marginal cumulative and appropriate number of the conditional cumulative distributions, is introduced. Here this procedure is tested on the nonlinear ion acoustic wave in a plasma. The numerical micro-solver is the one dimensional electrostatic particle-in-cell code. It is shown that particle correlations related to wave structures are better preserved by the new model. The lifting step is critically related to the noise in system. The enlarged noise, rise of correlations, trapping of particles during the wave steepening can seriously violate the basic assumptions of the equation-free approach.