I don't know that the precise average exists, but NIH posts a lot of
information about funding--by institute, by state, by award type, etc. at:

http://www.nih.gov/grants/award/award.htm

I've done several presentations and other reports lately and have used data
from this site. I know at least that you can obtain total R01 funding and
the number of R01 awards made, hence calculate the average. But, you can get
far more than that from this site...Bill

Bill Caskey, PhD
Director, Research & Grants Administration
Children's Mercy Hospitals & Clinics
2401 Gillham Road
Kansas City, MO  64108

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