The paper compares several methods for computing robust 1-α confidence intervals forσ12-σ22, orσ12/σ22, whereσ12andσ22are the population variances corresponding to two independent treatment groups. The emphasis is on a Box-Scheffe approach when distributions have different shapes, and so the results reported here have implications about comparing means. The main result is that for unequal sample sizes, a Box-Scheffe approach can be considerably less robust than indicated by past investigations. Several other procedures for comparing variances, not based on a Box-Scheffe approach, were also examined and found to be highly unsatisfactory although previously published papers found them to be robust when the distributions have identical shapes. Included is a new result on why the procedures examined here are not robust, and an illustration that increasingσ12-σ22can reduce power in certain situations. Constants needed to apply Dunnett’s robust comparison of means are included.