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The formula below will return the sum of all items in the range excluding the top 2.
=SUMIF($F$2:$F$8,"<" &LARGE($F$2:$F$8, 2),$F$2:$F$8)
When ranking, make sure that your data don't permit duplicates. LARGE($F$2:$F$8, 2) will return the second ranking value even if it is equal to the first and/or equal to the third. Therefore the above formula will return the wrong result if 2nd rank is not unique. You wouldn't have that problem with the formula below and you may prefer it for that reason.
=SUM($F$2:$F$8)-LARGE($F$2:$F$8, 1)-LARGE($F$2:$F$8, 2)
To break ties in ranking use the ROW() function in a helper column. Presume the values 1, 3, 2 and 3 in A1:A4 (all integers). In the helper column's row 1 write this formula = A1+ROW()/100 and copy down. The divisor must be a power of 10 which is larger than the number of rows in column A so that the result of ROW()/100 is always < 1. Now the helper columns shows
1.01
3.02
2.03 and
3.04
The two values of 3 in column A are no longer identical in the helper column. Second place is now decided by the location of the competitor in the list. You should use this system if the column to sum is not the column used for ranking.