This paper provides a new framework for analyzing white noise disturbances in linear systems; rather than the usual stochastic approach, noise signals are described as elements in sets, and their effect is analyzed from a worst-case perspective. The paper studies how these sets must be chosen to have adequate properties for system response in the worst-case, statistics consistent with the stochastic point of view, and simple descriptions that allow for tractable worst-case analysis, The method is demonstrated by considering its implications in two problems : rejection of white noise signals in the presence of system uncertainty and worst-case system identification.