When dealing with numerical variables, some of the most interesting and useful inferential problems concern estimating and testing for variances. Since the variance represents the population functional that measures the variability, that is the tendency of the variable of assuming data with a given degree of dispersion around the mean, these inferential problems are very important in many fields of application. Examples include: in statistical quality control and in performance analysis, reduction of waste and improvement of performance are related to lowering of variances; the efficiency of any estimator can be measured through its variance; and tests for the equality of variances are important for choosing the appropriate test statistic in the two-sample *t* test for mean comparisons, see Section 2.2.

In the presence of categorical data, unless it is possible and sensible to transform the original variables assigning suitable scores to the modalities, mean and variance, as a measure of central tendency and variability of a distribution, respectively, cannot be computed. In many problems, for example in the presence of nominal variables, score transformation is nonsense. Let us consider that several real phenomena may be represented only by nominal categorical variables and many others by ordinal variables, the score transformation is subjective and questionable because it may change the original information provided by data (e.g., ...

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