For a given positive random variable $V>0$ and a given $Z\sim N(0,1)$ independent of $V$, we compute the scalar $t_0$ such that the distance in the $L^2\left(ℝ\right)$ sense between $Z{V}^{1/2}$ and $Z\sqrt{t_0}$ is minimal. We also consider the same problem in several dimensions when $V$ is a random positive definite matrix.

The Goodman-Kruskal measure, which is a well-known measure of dependence for contingency tables, is generalized to the case when the variables of interest are categorized by linguistic terms rather than crisp sets. In addition, to test the hypothesis of independence in such contingency tables, a novel method of decision making is developed based on a concept of fuzzy $p$-value. The applicability of the proposed approach is explained using a numerical example.