Statistics for Process Control Engineers

27Pearson Distribution

Karl Pearson, perhaps most known for the R² coefficient used to assess correlations, developed a family of 12 distributions.^[18–21] His work predates the publication of almost all others, bar the normal distribution. Indeed, some well‐used distributions such as the F distribution, gamma distribution and Student t distribution are special cases of Pearson’s work. Pearson’s objective was that a distribution be fitted not only to the mean and variance of the data, but also to the skewness and kurtosis. Indeed, it was Pearson who originally defined skewness and kurtosis as additional measures of the character of a distribution. In principle, maximum likelihood estimates for each of the shape parameters can be then determined from the distribution’s moments. In practice this is mathematically complex. Instead, as we have done throughout this book, we apply the numerical curve fitting approach described in Chapter 9.

All the distributions are described only in PDF form. In most cases the CDF exists but is too complex to be used other than by specialist software products. Calculation of skewness and kurtosis is, for some of the distributions, similarly complex, in which case they have been omitted.

Figure 27.1 shows the range of skewness and kurtosis that can be represented by each of the distribution types. Skewness is plotted as γ² only so that the plots are linear or approximately so. This does not imply that, by taking the negative root, the distribution ...