Many waves exhibit characteristics that depend on time and/or frequency. For example, the frequencies of a pulse propagating in a dispersive medium travel at different velocities. This kind of dependency gives rise to the need for joint time- frequency analysis and methods for describing the local temporal and spectral nature of waves, particularly pulses and transients. Useful concepts arising for this description from time-frequency theory are the average frequency at each time of a wave, and the spread about that average. These quantities are obtained as conditional moments of a time-frequency density (TFD). We explore the conditional variances of some common TFDs, and determine when these are positive (which isn't always the case). We also investigate local characteristics of a wave, in terms of these conditional moments, and show through experimental results that they robustly characterize the time-frequency behavior of transients and pulses.