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We report on the analysis of virtual powder-diffraction patterns from serial femtosecond crystallography (SFX) data collected at an X-ray free-electron laser. Different approaches to binning and normalizing these patterns are discussed with respect to the microstructural characteristics which each highlights. Analysis of SFX data from a powder of Pr0.5Ca0.5MnO3 in this way finds evidence of other trace phases in its microstructure which was not detectable in a standard powder-diffraction measurement. Furthermore, a comparison between two virtual powder pattern integration strategies is shown to yield different diffraction peak broadening, indicating sensitivity to different types of microstrain. This paper is a first step in developing new data analysis methods for microstructure characterization from serial crystallography data.