Average Total Time specific to dates using likely sumproduct

I'm having some problems figuring out the formula that will take the average of a list of values that are within a date range. Example:
Date Total Time
1/1/2015 5
1/15/2015 10
2/1/2015 5
2/15/2015 10

And the table I want to produce would be:
Jan/1/2015 7.5
Feb/1/2015 7.5
Mar/1/2015 0

I want just one formula that computes the value 7.5 where that value is located in the 2nd table.

I know I can use a pivot table, and I'm even aware of how to do it. However I hate using pivot tables for a variety of reasons which is why I'm trying to generate it by hand.

I can only provide dummy data as my company has a non disclosure on me. Normally I would have just attached the sheet.

Lookup AVERAGEIFS function in help.

average_range is the entire "Total Time" column
criteria_range1 is the entire "Date" column
criteria1 is ">="&xx, where xx is the reference for the cell containing "Jan/1/2015" (needs to be Excel date serial encoded, not text)
criteria_range2 is the entire "Date" column
criteria2 is "<="&EOMONTH(xx,0)], where xx is the reference for the cell containing "Jan/1/2015" (needs to be Excel date serial encoded, not text)

Copy down.

Thanks that solved it. I'll figure out how to add solved to the thread. Surprised I got stuck on it honestly.