Science policy frequently hinges on reliable assessment of the uncertainty in predictions derived from various models. Because of systematic and random errors in the parameters of scientific models, measured values are commonly considered to be stochastic quantities following a Gaussian distribution, with the uncertainty characterized by the estimated standard deviation. We present analysis of multi-decade trends in the reported values of fundamental physical constants and projections of future energy demand and population growth, which reveals that a Gaussian, or normal, distribution grossly underestimates the frequency of events lying very far from the mean. The probability of large deviations is instead well fit by a simple exponential. This asymptotic behavior appears naturally in a compound distribution where both the mean and standard deviation are normally distributed stochastic quantities. We illustrate this formulation by estimating the probability of catastrophic sea-level rise resulting from global wanning.