Previous methods of estimating species richness from ecological data are problematic. Many assume that the data follow particular species distributions, most often meaning that underlying abundances are entirely even (e.g., the Chao 1 index). Any such estimator will provide a lower bound only, so it will be systematically wrong. Fits to more substantive theoretical abundance distributions can yield more realistic estimates. Globally distributed ecological data representing trees and terrestrial animals are fit with a stripped-down equation that combines two of the most basic distributions in statistics. This compound exponential-geometric series (CEGS) model predicts counts accurately, either when species inventories are split and the halves cross-tested or when inventory pairs are matched on their highest counts and cross-tested. Estimating richness, degrading the data, and recomputing richness shows that the method yields precise and accurate values. CEGS explains key patterns in nature in an intuitive and elegant way.