I have a population dataset in raster format (each pixel has a value showing number of people in that pixel) and a polygon shapefile with 10 districts. What I need, is to calculate population for each district. I thought it would be easy but I'm having difficulties... I use ArcGIS 10.3. I tried to turn raster to polygon, so that every square resembles a pixel. Then I tried to spatially join the two shapefiles, However, the numbers are wrong.
It sounds to me like the Zonal Statistics tool will work for you. I typically use Zonal Statistics as Table because it is really the tabular values that I want most (rather than a raster with a calculated value for each zone).
I think that the specific statistic you want is the SUM since you would add up the population from each cell into the entire zone.
Set up your polygon "Districts" layer as the "Input raster or feature zone data" input and your population raster as the "Input Value Raster". Then designate an output table to store the values. You could leave statistics set to "ALL" or change it to "SUM" if you are certain that is all you need.
Sometimes as a second step you may find it desireable to do a tabular join back to the Districts layer and your tabular output from Zonal Statistics as Table is set up to do that for you by using an OID for FID field in the output that matches your Zones (districts).