Here is how I would do that.
This procedure will give you a point shapefile with a point in each census centroid storing the mean value of the raster in that census.
First you need to store the X and Y coordinates of each census' centroid in your census polygon:
- open the Attribute Table of the census layer
- Add a field and call it (e.g.) X
- Use the Calculate Geometry on this field (right click on the name of the new field ni the Attribute Table and click "Calculate Geometry")
- Choose "X Coordinate of Centroid" and then OK
- Do the same for the Y, but here choose "Y Coordinate of Centroid"
Once you have the coordinate informations in the Attribute Table of your census polygons, you can extract them as a table by going in the Attribute Table options -> Export.
Now you have a table file representing only the attributes of you census layer. You can use the "Add XY Data" to create an Event Theme of the point centroids based on the XY coorindates you have just calculated:
- Go to File -> Add Data -> Add XY Data
- Choose the table you extracted and the X Y field in it you created and calculated previously (make sure to choose the right Spatial Reference here, it would normally be the same as your census data or the Data Frame of you map)
Now you should export the points you should see in your project as a new ShapeFile:
- Right click on the Event Theme representing the points added to the map after using the "Add XY Data" in the Table Of Contents
- Data -> Export Data and save it as a point ShapeFile
You can see the same procedure in more detail at this link.
Now that you have a point Shapefile representing the position of each census centroid, you need to assign it the mean value taken from the raster at each census zone.
Here I am assuming you have a Spatial Analyst license as you said you used the Zonal Statistics tool.
- Use the Zonal Statistics tool choosing MEAN as statistics_type parameter to produce a raster with as many cells as your input raster but with values of the mean of each census in it.
- Use the Extract Values to Points with a. your new centroid layer and the raster coming from the Zonal Statistics to create another point ShapeFile with the same points as in the centroid layer and the value of the mean found in the mean raster.