Variability orderings indicate that one probability distribution is more spread out or dispersed than another. Here variability orderings are considered that are preserved under conditioning on a common subset. One density f on the real line is said to be less than or equal to another, g, in uniform conditional variability order (UCVO) if the ratio f(x)/g(x) is unimodal with the model yielding a supremum, but f and g are not stochastically ordered. Since the unimodality is preserved under scalar multiplication, the associated conditional densities are ordered either by UCVO or by ordinary stochastic order. If f and g have equal means, then UCVO implies the standard variability ordering determined by the expectation of all convex functions. The UCVO property often can be easily checked by seeing if f(x)/g(x) is log-concave. This is illustrated in a comparison of open and closed queueing network models.

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn. Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.