Prompt gamma-ray spectra sometimes contain hundreds of characteristic peaks, and the masses of the sample components can be determined from dozens of gamma-ray peak areas, some of which are affected by spectral interferences. Reliable qualitative and quantitative analyses can only be performed by using a precisely calibrated detector system, an accurate spectroscopic data library, high-quality spectroscopy software, and a sophisticated method to convert raw data into chemical composition. This article describes the steps of the chemical analysis with prompt gamma activation analysis (PGAA). A complete data reduction method has been constructed that handles the large number of spectral peaks using statistical procedures, identifies the chemical components based on statistical criteria, and determines the sample composition with a least-squares fit of the masses calculated from the individual peaks. The calculation of the uncertainties as well as the detection and analytical limits are discussed in detail. The validation of the method is also presented. The method can also be used in the evaluation of the results of other spectroscopic techniques.