I have a raster from the TRMM 3B43 product that contains a grid of 0.25 x 0.25 tiles (in degrees). I also have a shapefile with a number of districts. I want to know how many raster tiles are included in each district (polygon). Ideally, I'd like to know it as a continuous measure (i.e. polygon A contains 2.43 tiles) but if it only counts complete tiles its also OK.
While reclassifying as mentioned above is one method, another is the Tabulate Area Tool.
This will allow you to tabulate the area of individual pixels based upon the polygons you wish to examine, and will also tell you the number of pixels per polygon. It is commonly used for automating an error matrix (with some additional minor calculations required).
Convert your raster into a binary value of 0 for no data and 1 for data
Zonal Statistics (spatial Analyst) or Zonal Statistics as Table (spatial Analyst)functions to calculate the number of grids within each of your polygons. Parametrize the function using SUM so that the zonal summary counts all the cells and sums the cells within the polygons. Since all cells with data are coded as 1 the sum will give you the number of cells per polygon. However, for smaller polygons you will still have 1 for each of the polygons, even if the same raster cell covers multiple small polygons. This will inflate the number of cells if you use the statistics from the zonal summary.