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# Twinkie Town Analytics Fundamentals: Using Linear Weights to Accurately Measure Run Production

Part 4: Run Value Stats: wRC, wRAA, Batting Runs, and wRC+

This is the fourth lesson in the series Twinkie Town Analytics Fundamentals. For more information on this series, what I hope to achieve by doing it, and the topics that will be covered please take a look at the series Introduction. If you have topics you’d like to be explored in this series, please leave a comment and let me know!

Previously:

Introduction

Part 1: The Flaws of Batting Average

Part 2: Busting the Myths of Pitcher Wins and ERA

Part 3: RBIs, A Misleading Statistic

I’ve begun each part of this series thus far with an example from Twins history that isn’t quite what it seems on the surface. These examples served to put in place some critical building blocks for understanding advanced baseball analytics – on base percentage, slugging percentage, weighted on base average, base out/states, and run expectancy. Today’s lesson will build on those concepts to introduce a handful of more complicated and nuanced statistics and the example I’ll be using is exactly what it seems on the surface. This is where we start to put some of these building blocks together to more comprehensively evaluate and measure offensive production.

Harmon Killebrew played 21 of his 22 seasons with the Twins franchise, beginning in 1954 as an 18-year-old rookie when the franchise was still located in Washington, D.C. and retiring in 1975 after a lone season with Kansas City. By almost any traditional offensive statistic you can put forward, the Hall of Famer ranks as the greatest offensive player in Twins history. He is the franchise leader in games played (2,329), slugging percentage (.514), total bases (4,026), home runs (559), runs batted in (1,540), bases on balls (1,505), and strikeouts (1,629). In many cases these numbers dwarf even the franchise’s second-best player’s numbers. Killebrew is the perfect example for today’s lesson because, as you’ll see, his achievements and run production – any way you want to slice them; traditional stats or advanced stats – stack up as the best in franchise history.

### Recap

In case you missed Part 3 where I introduced two key concepts that I’ll build upon today, let’s do a quick recap. Nearly all the advanced metrics to evaluate individual players in use today are based upon the idea of thinking of runs as fractions. We’ve been conditioned historically to think of runs as whole numbers, but the best approach to tackle questions about accounting for an individual player’s contributions (positive and negative) is to change our mindset and think about runs in smaller pieces. The two most basic elements of thinking about runs in fractions are base/out states and run expectancy.

When we think about game situations in baseball the first two pieces of data we need to know are 1) how many outs? And, 2) where are the base runners (if any)? These two pieces of data create the context in which the pitcher vs. batter matchup will occur. When we combine all possible combinations of base runners and outs you find that there are 24 possible base/out states. These range from bases empty and no outs to bases loaded and two outs. From the start of a half inning to the end of a half inning the game transitions between different base/out states as events happen.

From there, we can use historical play by play outcomes to calculate how many runs score from each base/out state on average. This average is called run expectancy and is often displayed in a table like below which shows the 2019 season results. Baseball Prospectus hosts historical versions of this table, here:

Ultimately this tells us the number of runs that can be expected to score, on average, from each combination of runners and outs. Runners on the corners with 2 outs resulted in 0.518 runs scoring by the end of the inning on average in 2019. Bases empty and no outs resulted in 0.544 runs on average.

This is very useful for evaluating individual players because we can use these figures to calculate how the result of a player’s plate appearance impacted the run expectation and use it as a measure of their productivity. These outcomes are the smaller pieces that add up to creating a run. This is the idea behind the metric RE24, a statistic that credits or debits batters and pitchers for their role in changing their team’s expectancy of scoring (or preventing runs) in a given inning.

### Extending the Concept: Linear Weights

Beyond simply accounting for the individual player’s run expectancy contributions, we might also be interested in understanding the run values of different events that happen. We want to know the true value of a single relative to a double, or to a walk. And we especially want to know the value of those things relative to making an out. This is the idea behind the concept linear weights.

We know intuitively a triple is more valuable than a single. But how much more? We might intuit, given the name of the event (triple) and the number of bases it is worth, that a triple is three times more valuable than a single. Using run expectancy and the real data we can test if that assumption is true (spoiler: it’s not). To do this, we can simply add the total run expectancy of all triples that occurred in our time period of interest – say, a single season – and then divide that total by the number of triples that occurred. This gives us the average run value of a triple. We can then repeat the exercise for singles and compare the two. We can also repeat the same exercise for all other outcome types – walks, doubles, home runs, even hit by pitches – and get a complete picture of the relative values of the events.

While this is the most accurate way to do it, it’s a little clunky to use. To make it more intuitive and user friendly, we can make some adjustments to make the run values relative to outs – instead of relative to average. I’ll spare you the nitty gritty math details and cut to the chase. The table below shows the adjusted weighted run values for different outcomes from the 2019 season.

Fangraphs hosts these weighted values on their Guts! Page and they are available for all seasons back to 1871. Returning to our question about triples and singles we find the average weighted run value of a triple (1.529) is only about 1.75 times more valuable than a single (0.87) – not three times like we might have assumed.

We use linear weights to properly measure and understand the values of different outcomes in relation to one another. These weighted values are the nuts and bolts of the gateway to advanced baseball statistics, weighted on base average (wOBA). In part 1 of this series I presented wOBA as the preferred comprehensive measure of individual offensive production. Not only is wOBA a better, more comprehensive measure than traditional stats, it is also the key input to other advanced metrics that help us answer different kinds of questions.

### Counting vs. Rate Statistics

There are two classes of metrics: counting stats and rate stats. Neither class is inherently more “right” or “wrong” than the other. Instead, think of them as different tools to be used to answer different questions.

You are likely very familiar with traditional stats belonging to both classes. Many of these have served as the language of the game for decades. Counting stats are those that accumulate over time. The more games played, the more plate appearances, the more opportunities a player receives, the more likely they are to have larger numbers of counting stats. Counting stats include things like runs scored, hits, home runs, and strikeouts, among many others. Rate stats are those that tell us how often something happens over some sample, most often as a ratio. Batting average, on base percentage, and earned run average are common rate stats and wOBA is a rate statistic.

When to use which class of stat depends in large part on the question we’re seeking to answer. When we’re looking for metrics to help us evaluate a player’s performance and put it into context, there are times when our questions are best answered by counting stats and other times when they are best answered by rate stats. Because of this there are both counting and rate based advanced metrics and they are all rooted in the linear weights concept and derived from wOBA.

In the remainder of this post, we’ll cover four advanced metrics (wRC, wRAA, Batting Runs, and wRC+) that are similar in many ways. Because the differences are nuanced and the metrics build on each other, I want to start off with a graphic showing how they fit together. I hope this will serve as a good orientation and make it easy for the rest of the content to be digested and understood.

### A Counting Version of wOBA: wRC

For a long time, baseball analysts used versions of a stat developed by Bill James in the 1970s called Runs Created that generally looked at a player’s times on base and bases advanced divided by his opportunities. In the years that followed this idea was modified several times by James and others, ultimately leading to Weighted Runs Created (wRC). This counting statistic is based on wOBA, making it able to more accurately value each outcome than prior versions of runs created. All we need to calculate wRC are the player’s wOBA and plate appearances; league average figures for wOBA, runs scored, and plate appearances; and an adjustment for scaling wOBA similarly as on base percentage. As a rough rule of thumb, 65 wRC in a single season is about league average. If we’re looking for a counting stat derived from the total value of a player’s outcomes this is where to look. For example, former Twins outfielder Larry Hisle posted a combined wOBA of .364 for his five season Twins career in the 1970s. One of Hisle’s (more famous) teammates, Tony Oliva, played his entire fifteen-year career with the Twins and posted a wOBA of .365. Were the two equally productive? On a rate basis, it would seem so. But we know the longevity of Oliva’s Twins career had value – those ten additional seasons played with the franchise should have provided considerable offensive value to the Twins – but how much? wRC is just the metric to tell us.

While the two players produced at roughly the same rate as measured by wOBA, the longevity of Oliva’s Twins career gives him a huge advantage in terms of cumulative production.

### Adjusting for Time Period: wRAA

A wOBA based counting stat like wRC, for all its goodness, does have some weaknesses. Most notably, it does not handle comparisons from different eras and leagues very well. The difficulty of creating runs in baseball varies over time for a myriad of reasons, including rule changes like lowering the mound (after 1968), modern medicine and training, and even the makeup of baseball itself (most recently 2018 and 2019). The net result of those kinds of variations is an overall run environment that is dynamic:

Since 1920, the number of runs scored per game by teams has varied between 3.42 (1968) and 5.55 (1930). Suppose we want to accurately compare Joe Mauer’s career production with Harmon Killebrew’s career production? What metric do we use to answer that question? Their careers took place in different run environments. In the 21 seasons Killebrew played between 1954 and 1975, the MLB average was 4.15 runs per game and ranged between 3.42 and 4.53. In Mauer’s 15 seasons between 2004 and 2018, the average was 4.49, with a low of 4.07 and a peak of 4.86. That might not seem like much difference on the surface but equates to a run environment in Mauer’s time that averaged 8% more than Killebrew’s.

As might be expected, the analytics community quickly realized some slight adjustments to wRC could give us a counting stat that accounts for the variations of historical era. Weighted Runs Above Average (wRAA) is based on wOBA and measures the number of runs a player contributes to their team compared to the average player. This comparison to average allows us to confidently compare players from different eras with a common benchmark. wRAA is calculated with essentially the same inputs and formula as wRC, plus a minor adjustment to scale it so zero is league average. A positive value denotes runs above average and a negative value denotes runs below average. As a rough rule of thumb, the MLB leaders in wRAA each season are in the 60s.

### Adjusting for Park Factors: Batting Runs and wRC+

Time period isn’t the only variation that impacts run creation. Baseball is unique in that the field of play is not uniform across the game. Park Dimensions, geography, and climate vary across the thirty current MLB stadiums and have changed often throughout history. Killebrew played his home games outdoors at Metropolitan Stadium. Mauer played the first six seasons of his career in the Metrodome, before the move to Target Field. And each of the road stadiums are different from one another. Should stats accumulated in different stadiums and playing dimensions be valued the same? How do we account for the run environments in different ballparks? These questions led to the development of linear weighted metrics (one counting and one rate) that include adjustments for park factors.

The counting stat is called Batting Runs and is essentially wRAA with an additional adjustment for park factors. Ultimately, batting runs are the core offensive component to Wins Above Replacement (WAR), which we will cover in more depth in a future part of this series. Like wRAA, the MLB leaders in Batting Runs each season are in the 60s.

wRC, wRAA, and Batting Runs are similar stats derived with linear weights and the latter two include adjustments for era and park differences. They are excellent stats, but they can be clunky to use – practically speaking, it’s not as intuitive for us to discuss these stats. We know what a good batting average or OBP looks like. We don’t as comfortably know that 35 wRAA or 80 wRC in a season is a strong year. Some of that is due to lack of familiarity and will get better with time as we all become more comfortable and versed in discussing these newer measures. But it’s always helpful to have one comprehensive number that is also intuitive to use.

That one statistic would be Weighted Runs Created Plus (wRC+) which is a rate statistic version of wRC that also includes park- and league-adjustments. It is scaled for league average to be at 100, meaning every point above or below 100 is a percentage point above league average. A wRC+ of 150 is fifty percent better than league average. A wRC+ of 90 is ten percent worse than league average. These adjustments make it a good stat to use for evaluating single seasons or careers in different eras, like our Mauer and Killebrew question previously. If you’re wondering, Killebrew was more productive by wRC+ 144 to 123.

### Summing Up

These stats are nuanced and only have some slight differences from one another. I realize it might be confusing to keep those details straight and understand what they are trying to tell us and how we should use them. I’ve put together this summary chart to make it easier.

To wrap up this lesson let’s go back to Harmon Killebrew. I listed his traditional stats in the introduction. He was dominant by those. Here are his numbers by the newer stats we’ve covered in this lesson:

In this series we’ve covered several different ways to measure offensive run production – traditional, advanced, rate, counting, adjusted for era, adjusted for parks – and Killebrew’s numbers clearly stack up as the best in franchise history any way you want to slice it, often by huge margins above the next closest player.

Now that we’ve completed our march through the advanced statistics for measuring offensive run production, Part 5 will turn our attention back to pitching and one of baseball’s most misguided stats – and the source of many poor managerial decisions – the save.

### Lesson Takeaways

• Thinking about runs in terms of fractions using the 24 base/out states and their associated run expectancies is the best tool we have to completely assess a player’s productivity and forms the foundation of sabermetrics
• The concept linear weights is built on run expectancy and gives us the ability to calculate and understand the true value (in terms of runs) of baseball outcomes.
• Linear weighted run values are calculated for the different outcomes (singles, triples, walks, etc.) and form the inputs for Weighted On Base Average (wOBA).
• wOBA is a gateway statistic that leads to more advanced stats that incorporate adjustments that allow us to confidently compare players across leagues, historical eras, and ballparks.
• Battings Runs is a counting statistic that includes adjustments for leagues and parks and serves as the batting input to the Wins Above Replacement (WAR) calculation.
• Weighted Runs Created Plus (wRC+) is a rate statistic that includes adjustments for leagues and parks and serves as the most helpful, easy to use, single statistic for comparing players.
• Harmon Killebrew was amazing and remains the most productive offensive player in franchise history.

Test Your Knowledge: Five Quiz Questions

#1: Linear weights are the fundamental components of the statistic wOBA. True or False?

#2: Counting statistics tell us how often something happens over a sample period. True or False?

#3: Weighted Runs Created (wRC) includes a league adjustment. True or False?

#4: Which statistic is the batting input to Wins Above Replacement?

#5: The adjustments made for batting runs and wRC+ to account for games being played in ballparks with different environments and dimensions are called ____________?

#1: True

#2: False

#3: False

#4: Batting Runs

#5: Park Factors

References:

The data sources are cited throughout this post. Like others who have tried to write and explain these subjects before, I relied significantly on the following resources:

• Book: Smart Baseball by Keith Law
• Book: The Book – Playing the Percentages in Baseball by Tom Tango, Mitchel Lichtman, and Andrew Dolphin
• Book: The Hidden Game of Baseball: A Revolutionary Approach to Baseball and Its Statistics by David Reuther, John Thorn and Pete Palmer
• Fangraphs’ indispensable library: library.fangaphs.com
• MLB’s glossary: mlb.com/glossary
• Baseball-reference.com

John is a contributor to Twinkie Town with an emphasis on analytics. He is a lifelong Twins fan and former college pitcher. You can follow him on Twitter @JohnFoley_21.