Variance upper bounds for discrete α-unimodal distributions defined on finite support are established. These bounds depend on the support and the unimodality index α. It is noted that the upper bounds increase as the unimodality index α increases. More information about the underlying distributions yields tighter upper bounds for the variance. A parameter-free Bernstein-type upper bound is derived for the probability that the sum S of n independent and identically distributed discrete α-unimodal random variables exceeds its mean E(S) by the positive value nt. The bound for P {S -nµ ≥ nt} depends on the range of the summands, the sample size n, the unimodality index α and the positive number t.