The expanded form of the Johnson system advanced in this paper can model any theoretically possible combination of the first four central moments of a random variable, i.e. any mean-variance-skewness-kurtosis combination exhibited by a yield, price or any other distribution that may be encountered in practice. None of the probability distribution models previously in the literature come close to achieving such property. An application involving Illinois farm-level corn yields is presented to illustrate the estimation, characteristics and use of the proposed system. Although the yield data analyzed is from the same state and crop, the skewness and kurtosis combinations implied by the best fitting non-normal models extend over a large region of the S-K plane, corresponding to both the SU and the SB families. Theoretically, it is known that several of these S-K combinations can not be accommodated by the most commonly used parametric models based on the Beta and the Gamma distributions, which corroborates the need for probability distribution models that can span larger areas of this space.