You should probably use rank statistics rather than averages. Assuming raw scores are positive and better if higher:
- normalized score= weight*(number of lower scores than the player)/(total number of scores-1).
0 if the player is the worst of them all, maximum if he is the best, half maximum for a median (not average) performance. Suitable for distributions that are known to be skewed and clustered, e.g. batting average in a season of baseball.
- normalized score= weight*(player score-minimum score)/(maximum score-minimum score)
0 if the player is the worst of them all, maximum if he is the best, half maximum at the midpoint between minimum and maximum raw scores. Suitable if raw scores are quite linear but their range is unpredictable, e.g. number of goals in a season of football or number of yards gained in a season of American football.