Certain distributions of relaxation times can be described in terms of a non-exponential response parameter, β, of value between 0 and 1. Both β and the relaxation time, τ0, of a material depend upon the probe used for studying its dynamics and the value of β is qualitatively related to the non-Arrhenius variation of viscosity and τ0. A solute adds to the diversity of an intermolecular environment and is therefore expected to reduce β, i.e., to increase the distribution and to change τ0. We argue that the calorimetric value βcal determined from the specific heat [Cp = T(dS/dT)p] data is a more appropriate measure of the distribution of relaxation times arising from configurational fluctuations than β determined from other properties, and report a study of βcal of two sets of binary mixtures, each containing a different molecule of ∼2 nm size. We find that βcal changes monotonically with the composition, i.e., solute molecules modify the nano-scale composition and may increase or decrease τ0, but do not always decrease βcal. (Plots of βcal against the composition do not show a minimum.) We also analyze the data from the literature, and find that (i) βcal of an orientationally disordered crystal is less than that of its liquid, (ii) βcal varies with the isomer's nature, and chiral centers in a molecule decrease βcal, and (iii) βcal decreases when a sample's thickness is decreased to the nm-scale. After examining the difference between βcal and β determined from other properties we discuss the consequences of our findings for theories of non-exponential response, and suggest that studies of βcal may be more revealing of structure-freezing than studies of the non-Arrhenius behavior. On the basis of previous reports that β → 1 for dielectric relaxation of liquids of centiPoise viscosity observed at GHz frequencies, we argue that its molecular mechanism is the same as that of the Johari-Goldstein (JG) relaxation. Its spectrum becomes broader on cooling and its unimodal distribution reversibly changes to a bimodal distribution, each of β < 1. Kinetic freezing of the slower modes of the bimodal distribution produces a glass. After this bifurcation, the faster, original relaxation persists as a weak JG relaxation at T → Tg, and in the glassy state.