To analyze two-way contingency tables with ordered categories, Iki and Tomizawa (2018) proposed a measure to distinguish two kinds of marginal asymmetry for the midpoint. However, that measure cannot necessarily judge all of the marginal asymmetries. We improve the measure and give a new measure for multi-way contingency tables. We also derive an approximate confidence interval for the measure.

We construct the multivariate probability distributions (Pólya, inverse Pólya, hypergeometric and negative hypergeometric) from the generalized quantum deformed algebras. Moreover, we derive the corresponding bivariate probability distributions and determine their properties ($\mathcal{R}\left(p,q\right)$-factorial moments and covariance). Besides, we deduce particular cases of probability distributions from the quantum algebras known in the literature.

In this article, we prove that induced modules of support $\tau$-tilting modules over blocks of finite groups satisfying inertial-invariant condition are also support $\tau$-tilting modules.