The aim of this paper is to obtain real bounds on the accumulated roundoff error due to the addition of positive independent random variables in floating-point arithmetic. By “ real bounds”, we mean an interval I such that the error belongs to I with a high probability. We show that the real bounds for a rather high probability (0.99 and more) are much tighter than those obtained from the known “strict” estimates. The length of the real distribution interval grows as fast as n 3 2 where n is the number of addends, i.e. n 1 2 times slower than what could be expected from the strict estimates. The method is extended to the process of numerical integration. Experimental results are given.