Because I know you visited ESPN.com this morning to do some math, here is a list of numbers for you: 0.29, 0.29, 0.16, 0.16, 0.15, 0.13, 0.13, 0.12, 0.1, 0.09, 0.07, 0.06, 0.05, 0.05, 0.05, 0.04, 0.04, 0.04, 0.04, 0.04, 0.03, 0.03, 0.02, 0.01. Per StatsPerform, that's the xG value of each of Liverpool's shots against Real Madrid in the Champions League final. The numbers are presented in descending order, from Mohamed Salah's dink toward the near post in the 16th minute (which was, like all of Liverpool's on-target efforts, parried away by Thibaut Courtois) to Jordan Henderson's 28-yard Y.O.L.O. shot off of a corner in the 41st.

There is a broad list of disclaimers about using xG in discussing a specific shot or match; xG is a lot more effective as a tool when used on aggregate over a longer period of time, and there's only so much you can learn from a tiny sample of data points. But if the numbers you're using are good, you can still learn things. Using it in a quick simulation, you find that Liverpool could have expected to score an average of 2.2 goals based on those shots. Their chances of scoring at least three goals (38%) were much higher than their chances of doing what they actually did: score none (9%).

This tells you that Liverpool was rather unfortunate in the final, something your eyeballs had probably already concluded. They took 24 shots to Real Madrid's four, and since Karim Benzema's disallowed goal from late in the first half didn't make it onto the score sheet, only one of the Blancos' shot attempts was particularly high quality.

Of course, that one shot was the match and tournament winner from Vinicius Jr. Fed by either a gorgeous low cross from Federico Valverde or a particularly poor shot -- I believing something different every time I see it (but at this exact moment, I'm thinking pass) -- Vini, who had found separation from defender Trent Alexander Arnold, got his feet right and tapped in a shot that was worth (a) 0.7 xG and (b) one actual goal. Real Madrid won, 1-0.

- MLS, LaLiga, Bundesliga on ESPN+: LIVE games, replays
- Soccer on ESPN+: FC Daily | Futbol Americas
- Don't have ESPN? Get instant access

There's no way to bend math in a way that suggests that, with those four shots, Real Madrid could have expected to win this match a majority of the time. But this was a nice experiment in the importance of particularly high-quality opportunities.

Again per StatsPerform, here's the xG value of Real Madrid's four official shot attempts: 0.70, 0.10, 0.06, 0.06. Because the likelihood of their best shot was so high, they could have expected to score at least one goal 75% of the time. That's less than Liverpool's 91%, but not as dramatic as "24 shots to four" suggests. If we create the same per-shot average from a different set of numbers, that 75% can drop quite a bit.

Let's say their four shots were worth 0.29 (the value of Liverpool's best shot), 0.23, 0.23 and 0.17 instead: Now their odds of scoring have dropped to 65%.

In soccer, a single, high-value scoring chance can make an outsized difference, especially in a knockout game in which talent levels are even and stakes are high. Over a long period of time, the tiny xG figures begin to add up and tell a story, but on a short-term basis, the big chances matter the most.

Quality over quantity

Consider the following game from earlier this season:

Villarreal vs. Bayern Munich, Champions League quarterfinals

- Total shots (180 minutes): Bayern 45 (3.03 xG), Villarreal 16 (2.28)
- Total shots worth 0.3 xG or more: Bayern 2 (0.64 xG), Villarreal 2 (1.04)
- Actual goals: Villarreal 2, Bayern 1

Generally speaking, getting outshot by a nearly 3-to-1 ratio is not a great way of going about your business and would lead to defeat more often than not. But against a prolific and relentless Bayern attack, Unai Emery's Villarreal focused all of its efforts on maintaining its defensive block, preventing good looks and hoping Bayern didn't connect on a semi-miraculous, low-percentage attempt. Most of Bayern's shots came from longer distances, and from an xG perspective, Villarreal actually created -- and converted -- the two best chances of the two-game set.

This is obviously not a new tactic, especially for an underdog in a knockout round. But in an era of heavy possession and ball dominance, it's almost refreshing to see that an old-school "defend and hit your marks in counter-attack" approach can still prosper.

In six knockout-round matches in the Champions League, Villarreal attempted 60 fewer shots, worth 1.7 fewer xG, than their opponents. But looking specifically at shots worth 0.3 xG or more -- for now, let's have fun and call them "big shots" -- they attempted eight and allowed just seven. That allowed them to blow past Juventus in the round of 16, upset Bayern and tie Liverpool after 150 minutes in the semifinals before the Reds finally pulled away.

(Perhaps Emery should have leaned more heavily on these principles during a disappointing LaLiga campaign, too: 12.9% of Villarreal's shots were "big shots," best in the league, but 10.8% of opponents' were as well, which was also a league-best mark. They leaned on control in the Champions League and thrived, while creating chaos in the league and finishing seventh.)

It was a similar story for Real Madrid: The Blancos attempted 63 fewer shots than opponents in the knockout rounds and generated 3.9 fewer xG, but they broke even on big shots and converted their opportunities better than opponents. It's easy to survey the late-round stats and assume that Real Madrid was a lucky champion, which might be true to a degree, but they significantly bridged any gaps with structure and big shots.

What made Real Madrid's Champions League run even more startling is that like Villarreal, they were a different team in league play, attempting far more shots than anyone else in LaLiga (0.21 per possession), but with only 8.7% of them qualifying as "big shots," eighth in the league and nearly equal to what opponents managed (7.8%). They won the shot quantity game far more than the shot quality game.

What works over a long league season is different than what works in knockout rounds. Carlo Ancelotti's squad pulled off different approaches and won two really big trophies.

The game's best volume shooters

While teams may benefit from different styles at different times, it probably goes without saying that the players who can most frequently generate high-quality chances are going to be rather valuable. Excluding penalties, here are the players who generated at least 15 "big shots" in league play this season:

• Robert Lewandowski, Bayern Munich (36 shots, 18 goals)

• Kylian Mbappe, PSG (22 shots, eight goals)

• Sadio Mane, Liverpool (21 shots, 10 goals)

• Patrik Schick, Bayer Leverkusen (20 shots, 12 goals)

• Christopher Nkunku, RB Leipzig (19 shots, 11 goals)

• Wissam Ben Yedder, Monaco (19 shots, nine goals)

• Son Heung-Min, Tottenham Hotspur (18 shots, eight goals)

• Tammy Abraham, Roma (18 shots, seven goals)

• Diogo Jota, Liverpool (17 shots, eight goals)

• Moussa Dembele, Lyon (17 shots, five goals)

• Raheem Sterling, Manchester City (16 shots, 10 goals)

• Mohamed Salah, Liverpool (16 shots, eight goals)

• Anthony Modeste, Koln (16 shots, eight goals)

• Marco Reus, Borussia Dortmund (15 shots, eight goals)

• Lautaro Martinez, Inter Milan (15 shots, eight goals)

• Randal Kolo Muani, Nantes (15 shots, four goals)

And here are a few more who deserve mention, both for decent volume and high-quality finishing.

• Juanmi, Real Betis (12 shots, 10 goals)

• Karl Toko-Ekambi, Lyon (13 shots, nine goals)

• Gaetan Laborde, Rennes (13 shots, nine goals)

• Pierre-Emerick Aubameyang, Barcelona/Arsenal (14 shots, nine goals)

• Dusan Vlahovic, Juventus/Fiorentina (11 shots, nine goals)

There are also a couple of names you may have expected to see here, but didn't: First, Real Madrid's Karim Benzema created 12 such shots and scored seven goals from them. That's not as prolific as you may have thought, but his league scoring total was plumped both by seven penalty goals (on 11 attempts) and the fact that he scored 13 goals from shots worth less than 0.3 xG.

Some were still pretty high quality -- his late winner against Valencia on Sept. 19, for instance, was worth 0.24 xG; other, more ridiculous strikes were worth far less.

Beautiful curled finish by Benzema ?? pic.twitter.com/ukL7wY9Z12

— ESPN (@espn) December 22, 2021

The xG on that one: 0.03.

Oh, and another name that didn't show up? Erling Haaland, though that one is probably easier to explain. One of the best in the game at putting away high-quality chances in the box, he generated 13 "big shots" in league play this season, scoring on eight of them. Had he not missed 10 league matches to injury, he would have easily shown up on the list above.

Meanwhile, 13 players created at least seven "big shots" in UEFA competitions:

• Sebastian Haller, Ajax (14 shots, eight goals)

• Abraham (12 shots, five goals)

• Guus Til, Feyenoord (11 shots, four goals)

• Arthur Cabral, Basel (10 shots, six goals)

• Lewandowski (nine shots, seven goals)

• Nkunku (nine shots, six goals)

• Cyriel Dessers, Feyenoord (nine shots, six goals)

• Salah (nine shots, four goals)

• Eran Zahavi, PSV Eindhoven (eight shots, four goals)

• Erling Haaland, Borussia Dortmund (seven shots, three goals)

• Mbappe (seven shots, two goals)

• Paulinho, Porto (seven shots, zero goals)

• Arkadiusz Milik, Marseille (seven shots, six goals)

Haller and Ajax generated a constant stream of high-quality looks in the Champions League group stage -- and that was only partially due to the fact that they played two matches against one of the most generous defenses in major soccer, that of Borussia Dortmund -- but they all disappeared in the round of 16.

In 180 minutes against Benfica, they attempted 27 shots but managed only two "big shots." Benfica matched that total and won the tie thanks in part to an unfortunate own goal from, of all players, Haller. He just couldn't stop scoring.