Many distributions result from manipulating data that follow another distribution. For example, if we collect data from a normal distribution, square each value and sum together a defined number of the squares, the results will follow a chi‐squared distribution. The control engineer might rightly be unconcerned by such a result. It is unlikely that process data would be manipulated in such a way. However, it highlights an important principle. While the raw data collected from a process might be normally distributed, any property derived from the data is likely not to be so.
For example, an inferential property is derived from independent variables. The form of calculation can influence the form of distribution. But most of the process variables we analyse are derived from others. A product composition, for example, depends on a number of independently set process parameters such as flows, temperatures and pressures. A process simulation would contain a complex calculation that derives composition from basic measurements. We can think of the process as this calculation.
In fact, by definition, a variable we wish to control must be dependent on others. So, while the independent variables might reasonably be expected to be normally distributed, the nonlinearity inherent to the process can significantly alter the form of distribution of the dependent variable.
We have seen that the Central Limit Theorem tells ...