If an account has shopping campaigns (they are assigned average position of 0), when I pull the account performance report, the average position is often less than 1.
What seems to be happening is that when calculating the avg position, the system does a weighted average over position and impressions, but it is also including shopping campaigns' 0 avg. position. this causes wrong avg. position to be reported.
To validate my theory, I tried the same thing in the ad words interface, and viola - wrong Average Position at the account level in the AdWords UI! (Avg. Position is less than Avg. Pos on Search or Display). I have more examples where avg. position actually drops below 1 due to this reason.

Probably the correct way to calculate the average is to leave out campaigns that have an average position of 0 when calculating the weighted average. Guidance from the AdWords team on how to read the Average Position stat would be helpful. An email informing me of a bug bounty award would be nice too :)
Cheers,
MG