These days it is commonly accepted in the financial world that the tails of an asset return distribution are much heavier than those of the Gaussian distribution. Furthermore the choice of distribution is crucial for suchpurposes as value-at-risk and option pricing. Unfortunately, opinion on the tail weight question is divided. One school of thought believes that they are most appropriately modelled by exponential distributions and another believes that power tailed distributions are needed. In the latter group, some believe that variances are finite and others believe that stable distributions should be used. The question that I will address is, given a competing family of distributions, whose tails are heavier than normal, how large a sample size n does one need to distinguish the tails of the distribution at a specified quantile p? Specifically, how large a sample do we need to distinguish the Laplace from the t distribution, and the t distribution from the stable distribution etc?