Wealth, earthquakes, blood vessels, turbulence and many other phenomena follow power-law probabilities with "black swan" behavior: extreme values such as billionaires or the 1906 San Francisco earthquake occur far more often than we would expect from seeing typical cases. The exponent of the power law determines the frequency of rare events and its measurement has been a challenge across fields. Power-law distributions arise from a symmetry, scale invariance, which is related to self-similarity. I introduce a new kind of power-law probability distribution that only assumes invariance to discrete scale transformations and thereby describes a wider variety of self-similar objects. I show how accounting for discreteness can resolve some of the difficulty of measuring power law exponents: popular maximum-likelihood inference methods are unstable in common empirical contexts and a discrete estimator rescues the method. This results in a 10% correction to measurements of the Gutenberg-Richter earthquake scaling constant and a new method for measuring scaling relationships in fractal objects like circulatory systems and Romanesco broccoli.