Abstract Extremes are rare and unexpected. This limits observations and constrains our knowledge on their predictability and behavior. Graphical tools are among the many methods developed to study extremes. A major weakness is that they rely on visual-inspection inferences which are subjective and make applications to large datasets time-consuming and impractical. Here, we advance a graphical method, the so-called Mean Excess Function (MEF), into an algorithmic procedure. MEF investigates the mean value of a variable over threshold, and thus, focuses on extremes. We formulate precise and easy-to-apply statistical tests, based on the MEF, to assess if observed data can be described by exponential or heavier tails. As a real-world example, we apply our method in 21,348 daily precipitation records from all over the globe. Results show that the exponential-tail hypothesis is rejected in 75.8% of the records indicating that heavy-tail distributions (alternative hypothesis) can better describe rainfall extremes. The spatial variation of the tail heaviness reveals that heavy tails prevail in regions of Australia and Eurasia, with a “hot spot” found in central Russia and Kazakhstan. We deem this study offers a new diagnostic tool in assessing the behavior of extremes, easy to apply in large databases, and for any variable of interest. Our results on precipitation extremes reinforce past findings and further highlight that exponential tails should be used with caution.