The statistics of liquid-to-crystal nucleation are measured rigorously by using a recently developed automated lag-time apparatus (ALTA). A single sample, in this case a sample of pure water both with and without an (insoluble) AgI crystal, is repeatedly cooled, nucleated, and thawed. Analysis of the data, coupled with a second kind of experiment, shows that the statistics of nucleation are consistent with a first-order kinetic mechanism over a wide range of supercooling temperatures. The limitations of classical nucleation theory are exhibited. Our analysis unifies many related experiments in biology, physics, chemistry, and chemical engineering.