For sports are head-to-head match-ups with scores, and have leagues with sufficient games per season, you can evaluate the amount that luck is involved by comparing the points-scored/points-allowed to win/loss over a season.
Merely empirically determine the appropriate exponent to input for the pythagorean win percentage[1]. Baseball uses an exponent slightly under 2, while basketball uses an exponent slightly under 14. The higher the exponent, the less likely a poor team is to defeat a good team.
In fact this can be generalized to sports without points if you have some other way of estimating a team/player's "quality" normalized to 1 being "average" which is all the points-scored/points-allowed ratio does.
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Note that the Jamesian (points-scored/points-allowed) quality metric doesn't work as well in games where draws are common (e.g. soccer/football), and different quality metrics will yield different exponents, so do not allow apples-to-apples comparisons. For what it's worth, applying James's quality metric to league football yields an exponent of about 1.3[2], which is in line with the intuition of ranking these 3 sports from most to least luck per win:
Soccer, Baseball, Basketball
1: https://en.wikipedia.org/wiki/Pythagorean_expectation
2: http://www.soccermetrics.net/soccer-pythagorean-theory/why-t...