This week on the **Wharton Moneyball Podcast**, the hosts were discussing World Cup results following the group play stage.

At one point, they talked about the variance in performance and the role that luck can play in winning or losing. They felt like they didn’t have as good an intuition about how variable the games were and one of the hosts, Eric Bradlow, said that he’d try and look into it and have an answer for next week’s episode.

The discussion reminded me of a chapter in one of my favorite books, **The Success Equation by Michael Mauboussin**. The book goes into nice detail about the role of skill and luck in sports, investing, and life. On page 78, Mauboussin provides the following equation:

**Skill = Observed Outcome – Luck**

From there, he explains the steps for determining the contribution of luck to winning in a variety of team sports. Basically, the amount of luck plays is represented as the ratio of the variance of luck to the variance of observed outcomes.

To calculate the variance of observed outcomes, we find the win% of each team in a given season and calculate the standard deviation of that win%. Squaring this value gives us the variance of observed outcomes.

When calculating the variance of luck we first find the average win% of all of the teams and, treating this as a binomial, we calculate the standard deviation as **sqrt((win% * (1 – win%)) / n_games)**, where n_games is the number of games in a season.

I scraped data for the previous 3 complete seasons for the Premier League, NHL, NBA, NFL, and MLB from **sports-reference.com. **All of the data and code for calculating the contribution of luck to winning for these sports is available on my **GitHub page**.

**Let’s walk through an example**

Since the guys on Wharton Moneyball Podcast were talking about Soccer, I’ll use the Premier League as an example.

First, we create a function that does all the calculations for us. All we need to do is obtain the relevant summary statistics to feed into the function and then let it do all the work.

league_perf <- function(avg_win_pct, obs_sd, luck_sd, league){
## convert standard deviations to variance
var_obs <- obs_sd^2
var_luck <- luck_sd^2
## calculate the variation due to skill
var_skill <- var_obs + var_luck
## calculate the contribution of luck to winning
luck_contribution <- var_luck / var_obs
## Results table
output <- tibble(
league = {{league}},
avg_win_pct = avg_win_pct,
luck_contribution = luck_contribution,
)
return(output)
}

Using the Premier League data set we calculate the games per season, league average win%, the observed standard deviation, and the luck standard deviation.

**NOTE:** *I’m not much of a soccer guy so I wasn’t sure how to handle draws. I read somewhere that they equate to about 1/3 of a win, so that is what I use here to credit a team for a draw.*

## get info for function
# NOTE: Treating Draws as 1/3 of a win, since that reflects the points the team is awarded in the standings
premier_lg %>%
select(Squad, MP, W, D) %>%
mutate(win_pct = (W + 1/3*D) / MP) %>%
summarize(games = max(MP),
avg_win_pct = mean(win_pct),
obs_sd = sd(win_pct),
luck_sd = sqrt((avg_win_pct * (1 - avg_win_pct)) / games))
## Run the function
premier_lg_output <- league_perf(avg_win_pct = 0.462,
obs_sd = 0.155,
luck_sd = 0.0809,
league = "Premier League")
premier_lg_output

Looking at our summary table, it appears that teams have had an average win% over the past 3 seasons of 46.2% (not quite 50%, as we will see in NBA, MLB, or NFL, since draws happen so frequently). The standard deviation of team win% was 15.5% (this is the observed standard deviation), while the luck standard deviation, 8.1%, is the binomial standard deviation using the average win percentage and 38 games in a season.

Feeding these values into the function created above we find that luck contributes approximately 27.2% to winning in the Premier League. In Mauboussin’s book he found that the contribution of luck to winning was 31% in the Premier League from 2007 – 2011. I don’t feel like the below value is too far off of that, though I don’t have a good sense for what sort of magnitude of change would be surprising. That said, the change could be due to improved skill in the Premier League (already one of the most skillful leagues in the world) or perhaps how he handled draws, which was not discussed in the book and may differ from the approach I took here.

**Let’s look at the table of results for all sports**

Again, you can get the code for calculating the contribution of luck to winning in these sports from my **GitHub page**, so no need to rehash it all here. Instead, let’s go directly to the results.

- NBA appears to be the most skill driven league with only about 15% of the contribution to winning coming from luck. Mauboussin found this value to be 12% from 2007 – 2011.
- The NFL appears to be drive most by luck, contributing 39% to winning. This is identical to what Mauboussin observed using NFL data from 2007 – 2011.
- The two most surprising outcomes here are the MLB and NHL. Mauboussin found the MLB to have a 34% contribution of luck (2007 – 2011) and the NHL a 53% contribution of luck (2008 – 2012). However, in my table below it would appear that the contribution of luck has decreased substantially in these two sports, using data from previous 3 seasons (
**NOTE:** throwing out the COVID year doesn’t alter this drastically enough to make it close to what Mauhoussin showed).

**Digging Deeper into MLB and NHL**

I pulled data from the exact same seasons that Mauboussin used in his book and obtained the following results.

- The results I observed for the MLB are identical to what Mauboussin had in his book (pg. 80)
- The results for the NFL are slightly off (47.7% compared to 53%) but this might have to do with how I am handling overtime wins and losses (which awards teams points in the standings), as I don’t know enough about hockey to determine what value to assign to them. Perhaps Mauboussin addressed these two outcomes in his calculations in a more specific way.

Additional hypotheses:

- Maybe skill has improved in hockey over the past decade and a half?
- Maybe a change in tactics (e.g., the shift) or strategy (e.g., hitters trying to hit more home runs instead of just making contact or pitchers trying to explicitly train to increase max velocity) has taken some of the luck events out of baseball and turned it into more of a zero-sum duel between the batter and pitcher, making game outcomes more dependent on the skill of both players?
- Maybe I have an error somewhere…
*let me know if you spot one!*

**Wrapping Up**

Although we marvel at the skill of the athletes we watch in their arena of competition, it’s important to recognize that luck plays a role in the outcomes, and for some sports it plays more of a role than others. But, luck is also what keeps us watching and makes things fun! Sometimes the ball bounces your way and you can steal a win! The one thing I always appreciate form the discussions on the **Wharton Moneyball Podcast **is that the hosts don’t shy away from explaining things like luck, randomness, variance, regression to the mean, and weighting observed outcomes with prior knowledge. This way of thinking isn’t just about sport, it’s about life. In this sense, we may consider sport to be the vehicle through which these hosts are teaching us about our world.