Try the Mean Center tool
You can input a Weight Field to the Mean Center tool -- is this what you mean?
For more information, take a look at the "how it works" doc.
Ah... that geometric mean!
Not that I know of. I'll have to do some checking. For sure Dissolve doesn't have the option.
Hi Amy - Did you ever find a way to do this in ArcMap?
Hi! We ended up using Python scripting outside of ArcMap and arcpy.