We present measurements of the whole scattering matrix as a function of the scattering angle at a wavelength of 632.8 nm in the scattering angle range 3¿--174¿ of randomly oriented particles taken from seven samples of volcanic ashes corresponding to four different volcanic eruptions: the 18 May 1980 Mount St. Helens eruption, the 1989--1990 Redoubt eruption, and the 18 August and 17 September 1992 Mount Spurr eruptions. The samples were collected at different distances from the vent. The samples studied contain large mass fractions of fine particles and were chosen to represent ash that could remain in the atmosphere for at least hours or days. They include fine ashfall samples that fell at a variety of distances from the volcano and pyroclastic flows that retained their fine fractions. Together, they represent a range of ashes likely to remain in the atmosphere in volcanic clouds following eruptions from convergent plate boundary volcanoes, Earth's most important group of explosive sources of ash. All measured scattering matrix elements are confined to rather limited domains when plotted as functions of the scattering angle following the general trends presented by irregular mineral particles. This similarity in the scattering behavior justifies the construction of an average scattering matrix for volcanic ash particles as a function of the scattering angle. To facilitate the use of the average scattering matrix for multiple-scattering calculations with polarization included, we present a synthetic scattering matrix based on the average scattering matrix for volcanic ashes and the assumption that the diffraction forward scattering peak is the same for randomly oriented nonspherical particles and projected-surface-area-equivalent spheres. This synthetic scattering matrix is normalized so that the average of its 1-1 element over all directions equals unity. It is available in the full range from 0¿ to 180¿ and can be used, for example, for interpretation of remote-sensing data after a volcanic eruption when the actual properties of the volcanic ash are not known. The measured results for the Mount St. Helens sample have been compared with results of Lorenz-Mie calculations for projected-surface-area-equivalent spheres with the refractive index of the Mount St. Helens particles. We find strong differences between measured and calculated values.