<p>The distribution of interval times between recurrent discrete events, such as Solar and stellar flares, reflects their underlying dynamics. Log-normal functions provide good fits to the interval time distributions of many recurrent astronomical events. The width of the fit is a dimensionless parameter that characterizes its underlying dynamics, in analogy to the critical exponents of renormalization group theory. If the distribution of event strengths is a power law, as it often is over a wide range, then the width of the log-normal is independent of the detector sensitivity in that range, making it a robust metric. Analyzing two catalogues of Solar flares over periods ranging from 46 days to 37 years, we find that the widths of log-normal fits to the intervals between flares are wider than those of shot noise, indicating memory in the underlying dynamics even over a time much shorter than the Solar cycle. In contrast, the statistics of flare stars are consistent with shot noise (no memory). We suggest that this is a consequence of the production of Solar flares in localized transient active regions with varying mean flare rate, but that the very energetic flares of flare stars result from global magnetic rearrangement that reinitializes their magnetohydrodynamic turbulence.</p>