Improved approximations to scaling relationships for species, populations, and ecosystems across latitudinal and elevational gradients.

Historically, allometric equations relate organismal traits, such as metabolic rate, individual growth rate, and lifespan, to body mass. Similarly, Boltzmann or Q(10) factors are used to relate many organismal traits to body temperature. Allometric equations and Boltzmann factors are being applied increasingly to higher levels of biological organization in an attempt to describe aggregate properties of populations and ecosystems. They have been used previously for studies that analyse scaling relationships between populations and across latitudinal gradients. For these kinds of applications, it is crucial to be aware of the "fallacy of the averages", and it is often problematic or incorrect to simply substitute the average body mass or temperature for an entire population or ecosystem into allometric equations. We derive improved approximations to allometric equations and Boltzmann factors in terms of the central moments of body size and temperature, and we provide tests for the accuracy of these approximations. This framework is necessary for interpreting the predictions of scaling theories for large-scale systems and grants insight into which characteristics of a given distribution are important. These approximations and tests are applied to data for body size for several taxonomic groups, including groups with multiple species, and to data for temperature at locations of varying latitude, corresponding to ectothermic body temperatures. Based on these results, the accuracy and utility of these approximations as applied to biological systems are assessed. We conclude that approximations to allometric equations at the species level are extremely accurate. However, for systems with a large range in body size, evaluating the skewness and kurtosis is often necessary, so it may be advantageous to calculate the exact form for the averaged scaling relationships instead. Moreover, the improved approximation for the Boltzmann factor, which uses the average and standard deviation of temperature, is quite accurate and represents a significant improvement over previous approximations.