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.
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.