Sharp Polynomial Upper Bound on the Variance

This short paper presents a new idea leading to a very sharp upper bound on the variance of interval-valued data. The computation of this sharp bound can be carried out in polynomial time in the worst case. For a whole class of interval data this bound is exact as it can be shown that it coincides with the maximum. The algorithm derives from posing the optimisation problem in probabilistic terms, i.e. thinking beyond the deterministic interpretation of an interval. Interval-valued variance can be seen from an imprecise probability's perspective. There are two alternative and non-competing connotations of an interval: a set of real values, and a credal set of all possible probability measures in the given interval. These two connotations do not precipitate any quantitative discrepancies for interval arithmetic. In fact, interval arithmetic provides a means to compute with these imprecise probabilistic objects. This work demonstrates the computational advantage that originates from looking at intervals from an imprecise probabilistic angle, and it may serve as a testimony towards filling the computational void that has for too long discouraged practitioners from computing with interval statistics.