The Johnson transformation optimally selects one of three families of distribution to transform the data to follow a normal distribution.
|Johnson family||Transformation function||Range|
|SB||γ + η ln [(x – ε) / (λ + ε – x)]||η, λ > 0, –∞ < γ < ∞ , –∞ < ε < ∞, ε < x < ε + λ|
|SL||γ + η ln (x – ε)||η > 0, –∞ < γ < ∞, –∞ < ε < ∞, ε < x|
|SU|| γ + η Sinh–1 [(x – ε) / λ] , where
Sinh–1(x) = ln [x + sqrt (1 + x2)]
|η, λ > 0, –∞ < γ < ∞, –∞ < ε < ∞, –∞ < x < ∞|
The algorithm uses the following procedure:
|SB||The Johnson family distribution with the variable bounded (B)|
|SL||The Johnson family distribution with the variable lognormal (L)|
|SU||The Johnson family distribution with the variable unbounded (U)|
For more information on the Johnson transformation, see Chou, et al.1 Minitab replaces the Shapiro-Wilks normality test used in that text with the Anderson-Darling test.
For information on the probability plot, percentiles, and their confidence intervals, go to Methods and formulas for distributions in Individual Distribution Identification.