You are confusing yourself by thinking that the NSIDC monthly extent is an average. It isn't. Its the monthly extent. NSIDC say things like "Arctic sea ice extent for May 2013 was 13.10 million square kilometers". No mention of "average" there. Its the extent for May, not the average daily extent during May.
Extent varies depending on the grid scale used, both in space and time. Daily extent isn't an average of hourly extent, and monthly extent isn't an average of daily extent.
A year ago I was confusing myself by thinking that the NSIDC monthly extent was an average of the daily values (and I believe that I am not the only one making that mistake). Now I am maybe wrong by using the term “Official NSIDC monthly average” and I should use the term “Official NSIDC Monthly Extent”. But anyway, what I am looking to highlight is precisely the difference between the “Official NSIDC Monthly Extent” and an average of NSIDC daily values on a given month.
NSIDC defines a daily extent as grids that have 15% or more of ice in a given day. If this grid has 15% or more of ice, then the grid counts as if it has 100% of ice. I believe that this criteria was good in the XX Century, but can overvalue the sea ice extent now, that we can have a lots of grids with 15% or a little more of ice. So I tend to be concern about the concept of daily extent.
But I find more misleading the concept of monthly extent. NSIDC defines monthly extent as the sum of grids that have 15% or more of ice in a month. But especially in July, October and November, the ice at the start of the month is very different of the ice at the end of the month. On July, the ice is melting quickly and on October and November, the water of the ocean is freezing quickly. So in July, the NSIDC monthly extent is close to the daily values that we have on July 5-12th. It doesn’t matter if afterwards we have a strong melt on the month. An in October and November, the NSIDC monthly extent is also close to the daily values that we have on October 18-25th and November 18-25th, respectively. Especially in October, it doesn’t matter if the first week of October we have values similar to those close to the minimum of the year and then we have a strong freeze-up. The Official NSIDC October value will reflect the strong freeze-up, not the slow start of freeze.
As a conclusion, what I mean is that if a daily extent could be overvalue now that the Arctic Sea Ice is melting, as never before in the history of the human being, the Official Monthly Extent is worst because it is overvalue against an average of the daily extent values.
So for me it is better to follow area or volume, instead of extent, and if I follow extent, I look at the daily values but I find very misleading the concept of monthly extent. And that is a concern when I see that a lot of models using monthly extent.