Abstract Expressions are derived for the envelopes which enclose various spatially averaged frequency response functions. The ultimate aim is to provide a scaling factor which may be applied to the results yielded by Statistical Energy Analysis, Asymptotic Modal Analysis, or Skudrzyk's mean-value method, to convert frequency averaged values to either maximum or minimum response estimates. It is shown that certain semi-empirical scaling factors which have been used previously may in fact be derived analytically by either modal or wave methods. These factors are found to yield reliable results for one-dimensional components such as beams or rods. For two-dimensional structures, such as plates, it is found that repeated natural frequencies (degeneracy) and irregular natural frequency spacing can have a significant effect on the response envelopes. By making use of the existing literature concerning statistical room acoustics, revised expressions for the response envelopes are derived which incorporate these effects, and good agreement with computed frequency response curves is demonstrated for rod, beam and plate elements.