A quiz team's average score over its first 8 rounds was 54 points. After the 9th round its average had risen to 55 points. By how many points did the 9th-round score exceed the team's average over the first 8 rounds?

The first 8 rounds gave $8\times 54=432$ points and the 9 rounds give $9\times 55=495$, so the 9th round scored 63. That is 63 - 54 = 9 points above the old average. 8 measures the excess over the NEW average (63 - 55) instead of the old one. 1 point is the rise in the average itself, which is what the extra 9 points becomes once it is shared out. 63 is the 9th-round score, not the excess.