A fortunate bounce that goes the way of your favorite team may tempt you to say: “it’s better to be lucky than good.” But are we even able to distinguish between the two? Advanced sports analytics begins to help us disentangle the effects of luck and skill.

So as to avoid the philosophical questions or semantics involved, let’s simply define luck as something that is not expected to continue – and teams experiencing a stretch of good or back fortune soon regressing to the mean – while skill persists over time. The law or large numbers tells us that, if we as much sports as we wanted, all lucky or unlucky deviations would “wash out” on average, and we would know the true skill of a player or team. Of course, we don’t live in such a data paradise. A simple way to quantify future win probability is with the Pythagorean expectation. At the most fundamental level, the true skill of a team depends on the number of points it scores and allows. It is possible that there exist some special skill in eking out close games, but winning games is likely to be simply a matter of “bunching” points to the best effect. For example, the 1960 World Series is famous in that the Pirates won even though “the losing team scored more than twice as many runs as the winning team, as the Yankees won three blowout games (16–3, 10–0, and 12–0), while the Pirates won four close games (6–4, 3–2, 5–2, and 10–9).” Analogies to the electoral college are obvious. The Pythagorean win expectation formula is:

where the exponent k depends on how much luck is involved. Larger values mean that the higher “quality” team wins more often. As I wrote in a previous post, the chances a game was won by the best team or the luckiest depends on the sport. The best fit exponents for different sports have been calculated:

- English Premiere League : 1.3
- NHL : 2.15
- NFL : 2.37
- NBA : 13.91

Surprisingly, the Pythagorean win expectation can better at predicting future win/loss record than even the past record. Better at predicting future record, even more than actual current record. *Compare this formula with the Hill equation in biology, which models cooperative behavior like the binding of oxygen to hemoglobin. *

Consider the 2016 MLS Cup. The Seattle Sounders prevailed in penalty kicks despite not generating a single shot on goal for 120 minutes of regulation and extra time combined. In contrast, Toronto had seven shots on goal, including one that looked like a sure game-winner, except for an incredible save by Stefan Frei. Had it gone in, everyone would have congratulated Toronto on a dominating 1-0 victory. Instead, Seattle ended up with the cup.

Or how about Leicester city, who overcame 5000:1 odds to win the English Premiere League last year? They benefited from poor showings from the traditional EPL powerhouses, and also were lucky enough to edge out quite a few close games.

While most expected a return to Earth after such a meteoric rise, I don’t think many expected such a fiery crash landing. This year, Leicester city is fighting to avoid relegation.

Ice hockey is (somewhat) better when I comes to rewarding the best team, but even then, games can be decided by a bounce of the puck. To help figure out if an NHL team’s success is attributable to luck or skill, we can turn to Corsi, Fenwick, and PDO.

Unlike baseball and football, hockey doesn’t have well defined “states” to analyze. Instead, we can use shots on goal as a way to approximate puck possession. Corsi is the sum of shots on goal, missed shots and blocked shots. Fenwick is the same with blocked shots excluded. Why are shots so important?

The basic idea is that generating scoring chances takes skill, but whether a goal is actually scored is unpredictable. Just putting the puck on net allows good things to happen, like a deflection or rebound, even if the original shot doesn’t go in. Also, your opponent can’t score you have the puck. So Corsi/Fenwick is a measure of skill independent of the “luck” of goals going in. Conversely, the sum of shooting percentage and save percentage is called PDO. The thought is that these values are the luck potion, that should tend to regress to 100% over time. Of course, PDO can remain high if you have an exceptionally good goalie, or sharpshooting skaters.