Upper bounds are derived for the probability that the sumSofnindependent random variables exceeds its meanESby a positive numbernt.It is assumed that the range of each summand ofSis bounded or bounded above. The bounds for Pr {S – ES≥nt} depend only on the endpoints of the ranges of the summands and the mean, or the mean and the variance ofS.These results are then used to obtain analogous inequalities for certain sums of dependent random variables such asUstatistics and the sum of a random sample without replacement from a finite population.