The fallacy of the average: on the ubiquity, utility and continuing novelty of Jensen's inequality

Biologists often cope with variation in physiological, environmental and ecological processes by measuring how living systems perform under average conditions. However, performance at average conditions is seldom equal to average performance across a range of conditions. This basic property of nonlinear averaging - known as 'Jensen's inequality' or 'the fallacy of the average' - has important implications for all of biology. For instance, a burgeoning awareness of Jensen's inequality has improved our ability to predict how plants and animals will respond to a warmer and more variable future climate. But for many biologists, the fallacy of the average is still a novel concept. Here, I highlight the importance of Jensen's inequality, provide a simple graphical approach to understanding its effects, and explore its consequences at atomic, molecular, organismal and ecological levels.