Calculating Spatially Weighted averages from a theissen polygon?

I have to create a theissen polygon with the spatially weighted averages for specific concentrations from different borings. I have the theissen polygons created but this only shows me the polygons based on the the point values not actually point concentrations that I would like. What tool can i use to get this info?

I only have ArcGIS free trial which limits the amount of tools I have access too. So i also have Qgis so being able to do that in either program.