This video demonstrates how to calculate and display centroids of polygons in ArcGIS 10.
With respect to the values generated in the video, our negative longitude values (X values) are approximately 110 to 120 degrees east of the Prime Meridian, which runs through Greenwich, England. Shanghai has latitude/longitude coordinates of 31.2 degrees North, and 121.5 degree East, so relative to the polygons exhibited when I first opened the attribute table and selected a few features, these centroid calculations are right on track.
For a nice table summarizing the qualitative scale at which a decimal degree measurement can be correctly identified, see the following Wikipedia entry for decimal degrees.
A summary of steps to calculate and display centroids as presented in the video:
0. Load a shapefile with polygon features as an active layer by right clicking on "Layers" in the "Table of Contents" pane and selecting "Add Data"
1. Open the attribute table for the shapefile that you've loaded as a layer by right clicking the layer name and selecting "Open Attribute Table".
2. Click the "Table Options" menu button in the top left corner of the attribute table window that was opened in Step 1, and choose "Add Field".
3. For the name of the field, type "centx" to represent the X coordinate of the centroid. Select the "Type" as "Double". Click "OK".
4. Repeat Step 2 and Step 3, but make the field name "centy" to represent the Y coordinate of the centroid. Select the "Type" as "Double", and click "OK".
5. In the Attribute Table, right click on the field header "centx" and select "Calculate Geometry".
6. For the property to be calculated, select "X Coordinate of Centroid", and select "Decimal Degrees" as the units, whatever unit you desire for your given projection. Click "OK".
7. Repeat Step 5 and Step 6, but choose the field "centy" on which to calculate geometry, and select "Y Coordinate of the Centroid" as the property to be calculated.
8. Click the top left "Table Options" menu button in the attribute table, and select "Export". Select "All Records" for the export, and click the folder icon next to the output table path to specify the type of file to be save. The format used in the video was "dBASE Table".
9. When asked if you'd like to add the new table to the current map, select "Yes".
10. Right click on the newly added table, and select "Display XY Data".
11. In the "Display XY Data" dialog box that appears, choose "centx" as the X Field and "centy" as the Y Field. Click "OK" when complete.
12. The centroids should now be drawn in a temporary event layer. If they are not drawn, make sure the layer is active with the box next to the event layer checked.
13. To save the collection of centroids as a shapefile, right click on the name of the temporary event layer and select "Data" then "Export Data". Make sure that "All features" are select under the Export drop down menu. Click "OK", and your centroids are now saved as a shapefile to your specified path.