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A simple but possibly insufficient solution is to use a formula. If we were to assume that you already had a list of unique names in the A column, you could fill another column with this:


If we need to create the list of unique names however, we're going to have to use a pivot table (or macros; or a really hacky formula solution).

Select the two columns, click Insert -> Pivot Table... -> Current Selection -> OK. Drag Name over in the Row Fields list, and drag Score over in the Data Fields list. Double-click on Sum - Score in the Data Fields list and select Average from the Function dialog.

(If this works for you, please click on the ✓ to accept the answer.)