The Guide to the Expression of Uncertainty in Measurement (GUM) defines standard measurement uncertainty as the standard deviation of a probability distribution that describes the uncertainty associated with an estimate of the measurand, and defines expanded uncertainty as a multiple of the standard uncertainty. Monte Carlo methods can produce the expanded uncertainty for 95% coverage as one half of the length of the interval whose endpoints are the 2.5th and 97.5th percentiles of the probability distribution of the estimate of the measurand (when this distribution is approximately symmetrical). This creates an opportunity for a paradox to arise: that the standard uncertainty, defined as a standard deviation, can be larger than the expanded uncertainty. We provide an example involving real measurement data where this paradox arises with high probability, and then offer a new definition of standard uncertainty that agrees numerically with the conventional definition in "normal" cases, but that is still reliable in "abnormal" cases.