We study the statistical properties of SIR epidemics in random networks, when an epidemic is defined as only those SIR propagations that reach or exceed a minimum size s c . Using percolation theory to calculate the average fractional size M SIR of an epidemic, we find that the strength of the spanning link percolation cluster P ∞ is an upper bound to M SIR . For small values of s c , P ∞ is no longer a good approximation, and the average fractional size has to be computed directly. We find that the choice of s c is generally (but not always) guided by the network structure and the value of T of the disease in question. If the goal is to always obtain P ∞ as the average epidemic size, one should choose s c to be the typical size of the largest percolation cluster at the critical percolation threshold for the transmissibility. We also study Q , the probability that an SIR propagation reaches the epidemic mass s c , and find that it is well characterized by percolation theory. We apply our results to real networks (DIMES and Tracerouter) to measure the consequences of the choice s c on predictions of average outcome sizes of computer failure epidemics.