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Cell Statistics in ArcGIS will calculate the mean (or statistic of choice) among multiple rasters on a cell-by-call basis.

Can QGIS do this and, if so, how?

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You can use the r.series GRASS command in QGIS to accomplish this.

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Since QGIS version 3.16, the cell statistics algorithm is be available as direct port of the ArcGIS cell statistics tool available. It is implemented in C++ for optimal performance and can cope with different sources of raster layers. For all the details you can visit the documentation.

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