Probability distributions, of widely differing shapes, that can be obtained by simple transformations of the standard normal distribution. The relationship to the normal distribution makes it easy to calculate, for example, a tail probability (see tail area). The entire family is defined bywhere α and β are constants, g is a function, Z is a standard normal random variable and Y is a random variable having a Johnson distribution.

There are three classes of Johnson distributions, referred to as the S_U, S_B, and S_L classes, which result from different choices for the function g(x). In the order given, for a variable x, the functions areThe last case gives a type of lognormal distribution.