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The #VALUE! error is likely happening because Excel is trying to calculate before all monthly values are available, or because one of the referenced cells contains text instead of a number.
The first part of your formula should check whether the required monthly cells are complete. Only after that should Excel run the calculation.
If your three monthly cells are H77, I77, and J77, you can use this:
=IF(COUNTBLANK('Metric Data Inputs'!H77:J77)>0,"N/A",IF(SUM('Metric Data Inputs'!H76:J76)=0,"N/A",SUM('Metric Data Inputs'!H77:J77)/SUM('Metric Data Inputs'!H76:J76)))
This will return N/A while any of the monthly cells are still blank. Once all three months have values, the formula will calculate the result.
Also, your formula includes J77, but you mentioned that H = June, I = July, and K = August. If August is actually in column K, then you should use H, I, and K instead of H, I, and J.
In that case, use this version:
=IF(COUNTBLANK('Metric Data Inputs'!H77)+COUNTBLANK('Metric Data Inputs'!I77)+COUNTBLANK('Metric Data Inputs'!K77)>0,"N/A",IF(SUM('Metric Data Inputs'!H76,'Metric Data Inputs'!I76,'Metric Data Inputs'!K76)=0,"N/A",SUM('Metric Data Inputs'!H77,'Metric Data Inputs'!I77,'Metric Data Inputs'!K77)/SUM('Metric Data Inputs'!H76,'Metric Data Inputs'!I76,'Metric Data Inputs'!K76)))
The main idea is to check for blank monthly values first, then check whether the denominator is zero, and only then perform the calculation.