To get the 5 month inventory turn over you would need the beginning balance of inventory for January, so you would need 6 terms in the average in your denominator. Also the sum of the monthly calculations would not equal the 5 month inventory turn over figure. You can see this if you try the sum in Excel.

Imagine you were doing the inventory turn over calculation over 24 months, and in one month, either the inventory was astronomically low or the COGS was extremely high, causing the turnover to increase to a level that could be considered an outlier. Imagine this took your turnover from 0.96 to 10. Perhaps a warehouse was burned down and all inventory was lost, etc. Now, over the course of the 24 months the lowered inventory (or increased COGS, same effect) for that particular month will not play as significant a role in the turnover ratio as they are being averaged over more periods.

Play around with the scenario in excel using the formulas, you what you suggested does not equal, except in the instance when COGS and Inventory are exactly uniform across all periods.