The paper introduces the concept of negatively associated (NA) random variables, which are defined as pairs of disjoint subsets of random variables where the covariance of increasing functions of these subsets is non-positive. The authors derive basic properties of NA random variables, including the closure under increasing functions of disjoint sets and the inequality \( P(X_i \leq x_i, i = 1, \cdots, k) \leq \prod_{i} P(X_i \leq x_i) \). They also show that negatively correlated normal random variables are NA and provide examples of NA distributions such as multinomial, convolution of unlike multinomials, multivariate hypergeometric, Dirichlet, and Dirichlet compound multinomial. The paper discusses the application of NA in sampling without replacement, multiple ranking and selection procedures, and categorical data analysis. It highlights that NA and positive association can coexist in models of categorical data, demonstrating the complexity and richness of negative dependence. The authors also compare NA with other types of negative dependence, showing that NA has unique advantages in terms of closure properties.The paper introduces the concept of negatively associated (NA) random variables, which are defined as pairs of disjoint subsets of random variables where the covariance of increasing functions of these subsets is non-positive. The authors derive basic properties of NA random variables, including the closure under increasing functions of disjoint sets and the inequality \( P(X_i \leq x_i, i = 1, \cdots, k) \leq \prod_{i} P(X_i \leq x_i) \). They also show that negatively correlated normal random variables are NA and provide examples of NA distributions such as multinomial, convolution of unlike multinomials, multivariate hypergeometric, Dirichlet, and Dirichlet compound multinomial. The paper discusses the application of NA in sampling without replacement, multiple ranking and selection procedures, and categorical data analysis. It highlights that NA and positive association can coexist in models of categorical data, demonstrating the complexity and richness of negative dependence. The authors also compare NA with other types of negative dependence, showing that NA has unique advantages in terms of closure properties.