Hi Bri,
I think this formuls follows you rules. I have assumed that dance
partner being judged are in column A, starting at A2, and the five
judge scores are in columns B to F. This formula could be pasted into
G2.

=ROUND(IF(COUNT(B2:F2)=4,(SUM(B2:F2)-(MIN(B2:F2)+MAX(B2:F2)))/(COUNT(B2:F2)-2),SUM(B2:F2)/COUNT(B2:F2)),2)

Ken Johnson