Context. Observations show that transverse oscillations commonly occur in solar coronal loops. The rapid damping of these waves has been attributed to resonant absorption. The oscillation characteristics carries information of the structuring of the corona. However, self-consistent seismological methods that extract information from individual oscillations are limited because there are fewer observables than unknown parameters in the model, and the problem is underdetermined. Furthermore, it has been shown that one-to-one comparisons of the observed scaling of period and damping times with wave damping theories are misleading. Aims. We aim to investigate whether seismological information can be gained from the observed scaling laws in a statistical sense. Methods. A statistical approach is used whereby scaling laws are produced by forward modelling using distributions of values for key loop cross-sectional structuring parameters. We study two types of observations: 1) transverse loops oscillations as seen mainly with TRACE and SDO and 2) running transverse waves seen with the Coronal Multichannel Polarimeter (CoMP). Results. We demonstrate that the observed period-damping time scaling law does provide information about the physical damping mechanism, if observations are collected from as wide range of periods as possible and a comparison with theory is performed in a statistical sense. The distribution of the ratio of damping time over period, i.e. the quality factor, has been derived analytically and fitted to the observations. A minimum value for the quality factor of 0.65 has been found. From this, a constraint linking the ranges of possible values for the density contrast and inhomogeneity layer thickness is obtained for transverse loop oscillations. If the layer thickness is not constrained, then the density contrast is at most equal to 3. For transverse waves seen by CoMP, it is found that the ratio of maximum to minimum values for these two parameters has to be less than 2.06; i.e., the sampled values for the layer thickness and Alfvén travel time come from a relatively narrow distribution. Conclusions. Now that more and more transverse loop oscillations have been analysed, a statistical approach to coronal seismology becomes possible. Using the observed data cloud, we have found restrictions to the loop's density contrast and inhomogeneity layer thickness. Surprisingly, for running waves, narrow distributions for loop parameters have been found.