Obtain normalized raster average per polygon?

I have a raster that contains percentages of total forest killed plus a polygon layer that defines different zones. My goal is to find a good statistical representative that describes the forest killed situation per zone. So what I did first is I used zonal statistics as table with the raster as the input raster and the zones as polygons as the feature zone. This calculated me an average per zone based on the raster cells in the zone. However, this destorts the result. One zone might only have 5 raster cells in there but all of these cells have a value of 50%, which gives me an overall average of 50% for this zone. Another zone might have 1000 raster cells and obtains also an overall average of 50%. So that doesn't show that the latter zone has more raster cells with fores killed. So I need to normalize it somehow. What I thought I could do is try and calculate a weighted average. So I would need to know how many raster cells there are in total and how many cells there are within each zone. With that I could divide the amount of raster cells per zone by the total amount of raster cells in the whole region and then times that by the average of the raster cells per zone?

Is that the right approach to not misrepresent the data? Also...how would I do that in ArcMap10??