Your objective is to average 12 previous periods and post the result, but only if the period is a multiple of 12. Here is a way to do that copying a single formula down the entire data range (no need to position the formula every 12 months).
I'm assuming your Yearly Average is column C and the data starts in row 2 (header in row 1). The period is column A. Enter this formula in C13:
Then copy the cell and paste it in C14 through the end of the data rows (you can select and navigate to the end with a few clicks or use the PageDown key to select the entire range).
How It Works
Every cell in Column C (starting with the first place you need it), gets the formula, which is what allows you to easily paste it in the entire range rather than in selected cells. For periods that are a multiple of 12, the formula calculates the average of the 12 periods ending with that one. If the period is not a multiple of 12, it displays the cell as blank.