In addition to this, I have a vector layer of the provinces of Vietnam: For each province I have a unique id and the name of the province. Lastly, I have an Excel spreadsheet containing for each province, the waste production rate per person. I'd like to group the population data for each province, a bit like "join attributes by location" with a vector, but I'm unsure I can do this with raster files. I'm starting to use raster info; I've looked around but haven't found anything helpful.
I would actually do this in reverse - create a raster from the vector provinces map, making sure you have aligned the rasters exactly, so each cell is associated with both a single province value and a single population density value.
Once you have accomplished this, you can run the GRASS program r.univar which will give you statistics by the province cells, including the sum of the population density for each province, which will be the grouping you need.
You can then put these values into a spreadsheet, divide by your waste column, and then join that spreadsheet back to the original vector layer to create your visualisation.
Please note I haven't tried this myself and don't know if r.univar will be available in the GRASS QGIS plugin, but I hope you find the workflow useful.