Find average of max values - mysql

Hi,

I have a routine that returns the max value of each group of records, roughly something as follows:

SELECT max (price) FROM product GROUP BY productcategory;

so, this will return the max price with respect to each product category.  I am actually interested in the average of such max prices.

I would think that there should just be a way to take the average of such results within a mysql statement, but that does not seem to be the case; instead, it seems as though I need to loop through records and calculate the sum and divide it by the recordcount.

Am I missing any obvious solutions?

Thanks,
Pete