The reason I found your initial description confusing is that you did not explicitly say that you have a formula and also a data validation dropdown list in the same cell. This is not unheard of but it's unusual and I'm wondering why you are doing it. Is there something you are not showing in this simplified example that allows the user to override the formula for some reason?
The problem is that if you refer to a blank cell, it is converted to zero by default. There are two possibilities I can think of:
Modify your formula to explicitly return a blank if it finds a blank. This is probably the safest and most straightforward. I have attached your sample file with this solution.
Formula:
=IF(Select_Complete="No","",LET(N,VLOOKUP(A16,Array_Data,4,FALSE),IF(N="","",N)))
Modify your formula to force results to be converted to text
Formula:
=IF(Select_Complete="No","",VLOOKUP(A16,Array_Data,4,FALSE)&"")
This might be an issue depending on what else you are doing with that result.
Go to Options > Advanced > Display options for this worksheet
and uncheck "Show a zero in cells that have zero values"
However, the cell will still have a zero value, will turn red from your CF, and may have other downstream side effects that you don't want (note my first paragraph). This is my least favorite.
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