2022 Blogmas Day 8 – Weighted Runs Above Average and Runs Created

Let’s dive back into the weeds. Hitting the ball and getting on base are important jobs for hitters. But what actually wins the game? Scoring more runs than the other team. Today, we’ll look at two ways of quantifying run production, both of which are weighted in the same way we saw with wOBA.

Runs Above Average
Weighted Runs Above Average (wRAA) roughly quantifies how many runs¹As a reminder, wOBA is calculated by considering the change in run-scoring probability per plate appearance, and thus describes how many runs each plate appearance is worth. a player contributes compared to the league average. It uses wOBA as its base for calculation, and operates as a counting statistic. So, two players with identical wOBA will only be differentiated in their wRAA by number of plate appearances. Explicitly, we have:

\text{wRAA} = \frac{\text{wOBA} - \text{League Avg. wOBA}}{\text{wOBA Factor}}\times\text{PA}.

Note that we divide by the wOBA Factor (or wOBA Scale), because it only exists to scale wOBA so that its average is the same as the league’s average OBP. The true run-change values that let us calculate wOBA were found prior to this final scaling process.

As a result, we just compare someone’s wOBA against the league average, which tells us how many more (or fewer) runs they’re worth per plate appearance, and multiply by plate appearances. Once you go through the trouble of calculating wOBA, you get wRAA more or less for free.

Since someone with a league-average wOBA would have a wRAA of exactly 0 (the numerator would be 0), league-average wRAA is also 0. Then, a positive wRAA means you contributed runs, while a negative wRAA means you were the cause of losing some runs.

Weighted Runs Created

We’ll eventually look at Weighted Runs Created Plus, wRC+, which combines the two things we’ve learned in our acronym soup: a “w” means we use the linear weights to properly calculate the relative values of different offensive results, and the “+” means we make adjustments based on other park factors and scale it so 100 is league average with a percentage point scaling system.

But first, let’s look at wRC, which is a counting statistic very similar to wRAA. In fact, it’s exactly the same, except we add in the average number of runs per player so that 0 is no longer the average. (Alternatively, wRAA is just wRC linearly scaled so that 0 is the average.)

\text{wRC} = \text{wRAA} + \frac{\text{Total League R}}{\text{Total League PA}} \times \text{PA}

Let’s say on average, players had 65 runs per 600 plate appearances. Then, a player with 600 plate appearances would have a wRC of wRAA + 65. It’s that simple.

To get wRC+, we take wRC per plate appearance and consider the same ballpark factors as OPS+, and scale so that 100 is league average and someone with, say, a 120 wRC+ has a 20% higher wRC rate than the league average.

This turns wRC (and by proxy, wRAA) from a counting statistic back into a rate statistic. Two players with the same number of plate appearances can be compared directly with wRC+, but if the number of plate appearances differ dramatically, you could say one player on average was better, but it’s possible that another player with a lower wRC+ had a higher overall impact on the season because they played more often. For that reason, you’d need to fall back to wRC or wRAA to fully compare total offensive impact.²Practically, it’s better to use wRAA for this comparison because having that league average of 0 simplifies everything. We use wRC because it makes getting to wRC+ simpler mathematically; nobody tends to use it on its own.

Over time, wRC+ has become a standard for quickly comparing offensive production, and due to that “+”, it’s very good for comparing across generations of players in particular. While OPS+ roughly looks at hitting ability (and feels more friendly because more casual fans can understand the calculation of OPS), its base statistic is poor; wRC+ is a shinier version with the same scale.

Let’s close this two-for-one post with some fun facts.

• In 2017, there were 7 teams whose collective wRC+ was 10% or more below average, while only one team (the Houston Astros) were more than 10% above average.
• Since I was born, the highest career wRC+ is 190, courtesy of Barry Bonds.
• Since I was born, the lowest career wRC+ among (non-pitcher) Minnesota Twins players is 70, by Denny Hocking. Among players who I’ve personally seen play, it’s the second-worst on the list, Alexi Casilla with a wRC+ of 73.³Not only have I seen him play, I watched him buy a baseball glove at my local Dick’s Sporting Goods.

Continue to Day 9 – Offensive Wins Above Replacement

• 1
  As a reminder, wOBA is calculated by considering the change in run-scoring probability per plate appearance, and thus describes how many runs each plate appearance is worth.
• 2
  Practically, it’s better to use wRAA for this comparison because having that league average of 0 simplifies everything. We use wRC because it makes getting to wRC+ simpler mathematically; nobody tends to use it on its own.
• 3
  Not only have I seen him play, I watched him buy a baseball glove at my local Dick’s Sporting Goods.