The relation between the mean difference and the standard deviation in continuous distribution models

The objective of the present work is to study the relations between the mean difference and the standard deviation with reference to the most common continuous theoretical distribution models. The continuous distribution models without shape parameters, those with only one shape parameter, and those with two shape parameters have been considered. The shape parameters encountered are inequality indexes, skewness indexes or kurtosis indexes. For the models without shape parameters the perfect equal ranking of the values of the two indexes have been verified. For the models with only one shape parameter it was seen that with variations in the shape parameter both indexes increase or decrease, so that the relation between them is growing. The ratio between the two indexes made it possible to determine the interval in which one index is greater than the other and the one in which it is less. Analogous results emerged for the models with two shape parameters, in particular the region in which one index is greater than the other and the complementary one. It was confirmed that for some models the mean difference has a wider field of definition in terms of the parameters than the standard deviation.