Here's how you can do averages, etc using the spatial database SpatiaLite. First import both of your layers into a spatialite DB. Let's call them "sensors" (the point layer) with a "pollution" column and "regions" (the polygons) with a "name" column. You can do this easily with the spatialite-gui, Import shapefile button.
Now this query will give you various statistics from the sensors for each polygon:
SELECT AVG(s.pollution) AS "Average Pollution",
MAX(s.pollution) AS "Maximum Pollution",
COUNT(s.*) AS "Number of Sensors",
r.name AS "Region Name"
FROM sensors AS s JOIN regions AS r
ON ST_Contains(r.geometry, s.geometry)
GROUP BY r.name
(Side note: If you have tens of thousands of points, you'll want to add a spatial index on the polygon table, and implement it in the query.)