Statistics for Process Control Engineers

33Asymmetrical Distributions

There are a wide range of continuous distributions that accommodate nonzero skewness. This chapter includes those that are not obvious members of the groups covered in the previous chapters. For the lack of any better way of sorting them they are largely in alphabetical order.

33.1 Benini

The three‐parameter Benini distribution is described by

(33.1)

(33.2)

The two‐parameter Benini distribution is obtained by setting α to 0.

Figure 33.1 shows the effect of varying α and β with x_min fixed at 0.1. Fitting Equation (33.2) to the NHV disturbance data gives x_min as 0.088, α very close to 0 and β as 0.164. With RSS at 0.1678 the fit is poor and the three‐parameter version offers no advantage over the two‐parameter one.

Figure 33.1 Benini: Effect of α and β on shape

Fitting to the C₄ in propane data gives 1.75 for x_min, 0 for α and 0.900 for β. RSS is 0.0865, making it one of the poorer choices.

For both versions, the calculations for mean and variance require mathematics beyond the scope of this book.

33.2 Birnbaum–Saunders

The Birnbaum–Saunders distribution is also known as the fatigue life distribution, indicating the purpose for which it was originally ...