There are four types of Johnson distribution. The most commonly referred to is the version known as SU – probably because it is unbounded. Not included here, for any of the distributions, are the very complex formulae for the calculation of skewness and kurtosis. If required, the calculations can realistically only be performed by a software product.
The first, known as Johnson SN, is the normal distribution but with additional shape parameters α and β. It is described by
It has no advantage over the classic normal distribution. As Figure 26.1 illustrates, as usual, α determines the location and β the dispersion. The modification that α and β make to the shape of the distribution can be exactly replicated by adjusting μ and σ in the classic normal distribution. Indeed, fitting all four parameters in Equation (26.1) will be problematic because there is not a unique set of values that minimises RSS.