Analysis of high-order correlations in fluorescence fluctuation spectroscopy was developed in the late 1980s but since then has been replaced by alternative brightness analysis methods. However, high-order correlation has important advantages in many experiments. We present a new cumulant-based formalism of high-order correlation that greatly simplifies data analysis. The new formalism is used to derive general expressions for variance of high-order correlations that show good agreement with experiment in a model system of fluorescently labeled DNA oligomers. A simulation of binary systems in which both diffusion time and brightness are varied illustrates clearly that high-order analysis has better sensitivity to brightness than fluorescence correlation spectroscopy (FCS). These results have implications for analysis of isomerization reactions and dual-beam FCS with flow. We also demonstrate that high-order correlations can detect photobleaching in the observation volume. The application of this formalism to many FCS-based experiments allows more accurate analysis in addition to describing more molecular parameters.