# 32Symmetrical Distributions

There remain a large number of distributions that are not extensions to those covered so far in this book. Further they bear little relationship to each other. This chapter includes those that are symmetrical, i.e. their skewness is zero. Asymmetric distributions follow in the next chapter.

The simplest example of a symmetrical distribution is the uniform distribution that we covered in Section 5.1. The most used is the normal distribution (Section 5.3). We have also covered others such as the Student *t* distribution (Section 12.5), logistic distribution (Section 22.1) and the Cauchy distribution (Section 27.4). This chapter groups together others that fall into this category. In the absence of any mathematical argument for arranging them in a particular sequence, they are presented largely alphabetically.

There are several situations where we might reasonably expect the skewness of process data to be zero. For example, when we check an inferential property against a laboratory result, it is equally probable that error will be positive or negative. The distribution of errors will have a skewness close to zero. Perhaps the most common application is in the analysis of process disturbances. Disturbances in one direction are likely to be equally probable as those in the opposite direction. Indeed, we will use the NHV data as the example to evaluate each of the distributions. Of the symmetrical distributions considered so far, the best choice is logistic ...

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