Johnson Curves

Johnson (1949) described a system of frequency curves that represents transformations of the standard normal curve (see Hahn and Shapiro, 1967, for details). By applying these transformations to a standard normal variable, a wide variety of non-normal distributions can be approximated, including distributions which are bounded on either one or both sides (e.g., U-shaped distributions). The advantage of this approach is that once a particular Johnson curve has been fit, the normal integral can be used to compute the expected percentage points under the respective curve. Methods for fitting Johnson curves, so as to approximate the first four moments of an empirical distribution, are described in detail in Hahn and Shapiro, 1967, pages 199-220; and Hill, Hill, and Holder, 1976. For a comparison between Pearson and Johnson distributions, refer to the Technical Notes section of Process Analysis.