Contrasts are sometimes used to compare a mixed effect. A common example can be the difference between two test scores — one at the beginning of the semester and one at its end. Note that we are not interested in one of these scores by itself, but only in the contrast (in this case — the difference). Since this is a linear combination of independent variables, its variance will match accordingly, as the sum of the variances. This "blending" of two variables into one might be useful in many cases such as ANOVA, regression, or even as descriptive statistics in its own right.
Another example would be comparing 5 standard treatments to a new treatment, hence giving each old treatment a weight of 1/5, and the new sixth treatment a weight of −1. If this new linear combination has a mean zero, this will mean that the old treatments are not different from the new on average. As before, linear combinations of random variables imply variance of sums equaling the sum of variance.