The observed evolution of the global-mean surface temperature over the twentieth century reflects the combined influences of natural variations and anthropogenic forcing, and it is a primary goal of climate models to represent both. In this study we isolate, compare, and remove the following natural signals in observations and in climate models: dynamically induced atmospheric variability, the El Niño-Southern Oscillation, and explosive volcanic eruptions. We make clear the significant model-to-model variability in estimates of the variance in global-mean temperature associated with these natural signals, especially associated with the El Niño-Southern Oscillation and explosive volcanic eruptions. When these natural signals are removed from time series of global-mean temperature, the statistical uncertainty in linear trends from 1950 to 2000 drops on average by about half. Hence, the results make much clearer than before where some model estimates of global warming significantly deviate from observations and where others do not.