# Advanced Hitter Statistics

Mike Trout is thought by many to be a sabermetric darling. His unique blend of skills, including patience at the plate along with speed and power in the field make him an excellent case study for learning some the advanced hitter statistics.

Gaining an understanding of the Basic Hitter Statistics enables quantification of the type and amount of value a player produces for his team. By using advanced metrics and statistics, further isolating the specific value a player is responsible for and what is due to the environment and circumstances around him becomes possible.

This article will serve to explain some of these advanced statistics, and how they are able to better articulate a player’s value compared to the basic statistics. This article will serve as a guide into the calculation and interpretation of some of the more telling hitter statistics. This will allow understanding and participation in the sabermetric community and in particular, at SaberBallBlog.

## On-Base Plus Slugging |OPS|

On-Base Plus Slugging, or OPS, is self-explanatory; one of the most straight-forward advanced statistics, OPS takes a batter’s two rawest abilities (getting on base, and hitting for power) and adds them together. As such, the calculation for OPS is:

$\large \textrm{OPS} = \textrm{OBP + SLG}$

OPS is on a scale of 0 to 5.000 (the interval goes from no hits/walks to all home runs), but the recent (2010-2015) average has been around 0.720. In the 2000s, it was closer to 0.760. As that generation of power hitters began to retire, and pitching and defense became more dominant, the average OPS decreased. It should be noted that while OBP and SLG aren’t true “percentages” despite being called such, OPS is definitely not a percentage. Some of the best batters in the league routinely have an OPS between 1.000 and 1.100, a mathematically impossible figure for probability-based percentages. In addition to this, OBP and SLG have different (although similar) denominators, and so they do not represent the same fractions.

While OPS is not mathematically rigorous, its value as a statistic lies in that it is very quick to calculate, interpret, and understand. For this reason it is one of the advanced statistics that is more popularly represented in the media. OPS also does a good job of helping quickly gauge a player’s offensive ability. It is simple for a batter to have either a good OBP or a good SLG. However, only the truly high-quality players in the league have both a good OBP and a good SLG, and this is reflected in their better than average OPS.

To interpret OPS, there is a handy chart, courtesy of Fangraphs:Note 1

Interpretation OPS
Excellent 1.000
Great .900
Above Average .800
Average .710
Below Average .670
Poor .600
Awful .570

Let’s use Mike Trout, centerfielder for the Los Angeles Angels of Anaheim as an example for calculation purposes. Trout had an OBP of .402 and a SLG of .590 (refer back to Basic Hitter Statistics to refresh on exact calculation). This produces:

$\textrm{OPS} = .402 + .590 = .992$

You can verify this on his stat pages here or here. Notice that sometimes there can be .01 point deviations due to rounding errors, but this is nothing to lose sleep over. As we can see from the chart above, notice that Trout performs at an excellent level, and is frequently one of the top players in the league each year.

## On-Base Plus Slugging Plus |OPS+|

Despite its awkward name, OPS+ is an even more useful version of OPS. Similar to ERA-, OPS+ takes a batter’s OPS and normalizes it on a scale to league average for the timeframe. Again, like ERA-, average is set at 100 and every point above/below represents 1% above/below the average. A batter with an OPS+ of 125 is 25% above the average OPS, and an OPS+ of 84 is 16% below the average OPS. The equation for OPS+ is:

$\textrm{OPS+} = \frac{\frac{\textrm{OBP}}{\textrm{OBP}_\textrm{lg}}+ \frac{\textrm{SLG}}{\textrm{SLG}_\textrm{lg}} - 1}{\textrm{C}_\textrm{PF}} \cdot 100$

OPS+ is a better metric than OPS because it controls for park factors ($\textrm{C}_\textrm{PF}$) and season-to-season league effects. With these added in, it becomes possible to compare the OPS of any player from any time period and see how they performed relative to their peers. This makes OPS+ a very good metric to immediately check a player’s productivity, and certainly better than more simple statistics such as batting average, HRs, RBIs, etc.

OPS+ is not the optimal batter metric, however, as it has its shortcomings. The main one is that it equally weights on-base points and slugging points, when there is evidence to the contrary. Note 2

Let’s examine Mike Trout again, relative to his peers. The American League average OBP was .318 and the average SLG was .412. The PF at the Angels’ home stadium was 94 for 2015, so Trout’s OPS+ is:

$\textrm{OPS+} = \frac{\frac{.402}{.318} + \frac{.590}{.412} - 1}{0.94} \cdot 100 = 180$

This indicates that Mike Trout had an OPS approximately 80% above the league average! This is very impressive.

However, another problem with OPS+ becomes apparent! This number is actually incorrect if compared to the number listed at Baseball-Reference, 176. There are a few reasons for this and correcting them can be messy, especially for those new to advanced statistics.

The inaccuracy of the calculation has the implementation of the park factor to blame. When using the park factor, the park in which Trout was playing at the times of the walks, hits, etc. must be used. Because the calculation above only uses the park factor for Trout’s home stadium, we only capture the stadium effect of half of his games accurately! The other half are played in other stadia with higher or lower park factors. For this reason, OPS+ is normally tabulated in a spreadsheet rather than by hand as there are many equations to perform before you have a final number. Building databases and spreadsheets that perform calculations much quicker will come with comfort in this method and experience.

## Weighted On-Base Average |wOBA|

Weighted On-Base Average is another statistic that does exactly what it sounds like! While OBP indicates how often a batter reaches base, not all bases are worth the same amount to the team. By giving each type of base (unintentional BBs, HBP, 1B, 2B, 3B, and HR) a “weight”, we are able to more accurately value this effect. This addresses the concern with OPS and OPS+, delivering a more complete metric. The weight is a decimal number that relates the value of one stat to the others. The decimals change every year, but in general the formula is something like the one below:

$\textrm{wOBA} = \frac{0.69 \cdot \textrm{BB}_\textrm{u} + 0.72 \cdot \textrm{HBP} + 0.89 \cdot \textrm{1B} + 1.27 \cdot \textrm{2B} + 1.62 \cdot \textrm{3B} + 2.10 \cdot \textrm{HR}}{\textrm{AB + BB - IBB + SF + HBP}}$

From the calculation, the bases mentioned above are multiplied by their weights and then divided by a modified type of plate appearances. The weights describe the value of one base to another. For instance, a home run is worth almost exactly 3 times an unintentional walk, per the wOBA formula ($\frac{\textrm{wOBA}_\textrm{HR}}{\textrm{wOBA}_\textrm{BB}} = \frac{2.10}{0.69} \approx 3$). The weights vary from season to season and are designed to produce a league average wOBA that is equal to the league average OBP. As such, the average wOBA each year is somewhere around 0.320.

As mentioned, these weights serve to remove the shortcomings of OPS and OPS+. wOBA is better than OPS and OPS+ because it paints a more accurate picture of a batter’s output. In addition to this, it is simple to use wOBA to convert into predicted run production, which in turn allows us to predict win value. This process will be examined further below.

For example, we will use our friend Mike’s 2015 stats to calculate wOBA. In 2015, Mike had 92 walks (14 of which were intentional), 10 HBP, 93 1Bs, 32 2Bs, 6 3Bs, and 41 HRs in 575 ABs with 5 SFs. His wOBA then, was:

$\textrm{wOBA} = \frac{(0.69 \cdot (92 - 14)) + (0.72 \cdot 10) + (0.89 \cdot 93) + (1.27 \cdot 32) + (1.62 \cdot 6) + (2.10 \cdot 41)}{(575 + 92 - 14 + 5 + 10)} = .420$

Because wOBA is read on an OBP scale, and league average wOBA was around .318 in 2015, this means Trout had a wOBA almost 100 points above average! This only further indicates how well he played. It is important to note, however, that if checking his page at Fangraphs, it will list his wOBA as .415. This is because the weights change year-to-year as mentioned. The weights for 2015 were all slightly below the ones we listed above, but those were only to serve as an average for purpose of example. The average weights exist mostly to help us calculate wOBA in-season, before knowing the true weights for the entire year. As an exercise, use the final weights for the 2015 season found here and see if you can reproduce the same number that Fangraphs lists.

## Weighted Runs Above Average |wRAA|

As indicated with wOBA, it is possible to turn the weighted bases into a run value prediction. To do this, use a player’s wOBA, the average wOBA for his league, a constant that normalizes wOBA based on the time period, and the plate appearances the player made. The equation looks like this:

$\textrm{wRAA} = \frac{\textrm{wOBA} - \textrm{wOBA}_{\textrm{lg}}}{\textrm{C}_{\textrm{wOBA}}} \cdot \textrm{PA}$

Again $\textrm{C}_{\textrm{wOBA}}$ is referred to as the “wOBA scale”. This is a constant that allows us to judge the run expectation produced given the current run environment in a league at the timeNote 3. Generally, the number is somewhere between 1.15 and 1.25. The wOBA scale number should value each point deviation from the league average wOBA as being worth approximately 5 wRAA over a typical, full 600 PA season.

To see how to calculate wRAA with actual season statistics, we will use Mike Trout’s 2015 numbers to once again show his prowess. From above, we know that his 2015 season wOBA was .415, the league average wOBA was .313, the wOBA scale was 1.251 and he had 682 plate appearances. Plugging in, we get:

$\textrm{wRAA} = \frac{.415 - .313}{1.251} \cdot 682 = 55.6$

So, Trout was worth 55.6 weighted runs above the average league player in 2015! If you look at his Fangraphs page, this number will vary slightly from the listed 55.4, due to rounding errors that we have propagated from above. If you are computing all of this in a spreadsheet, the numbers will appear appropriately.

## Weighted Runs Created |wRC|

Weighted Runs Created is an attempt to create a single statistic that measures a batter’s entire stat line (H, 2B, 3B, HR, SB, etc.) and delivers the value of that player. wRC explicitly uses the first part of the wRAA equation, but then adjusts for the league average R/PA as well, as below:

$\textrm{wRC} = (\frac{\textrm{wOBA} - \textrm{wOBA}_{\textrm{lg}}}{\textrm{C}_{\textrm{wOBA}}} + \frac{\textrm{R}}{\textrm{PA}} {\small\textrm{lg}}) \cdot \textrm{PA}$

Examining the equation, wRC is incredibly similar to wRAA. The main difference between wRC and wRAA is that the average is not set at 0 due to the league average R/PA. Using Trout’s 2015 stats again and the MLB average of 0.113 R/PA:

$\textrm{wRC} = (\frac{.415 - .313}{1.251} + .113) \cdot 682 = 132.67$

The true value from this metric, however is derived from our ability to adjust it for league and park factors, as we will see with wRC+.

## Weighted Runs Created Plus |wRC+|

Weighted Runs Created Plus is arguably the single greatest metric to measure a batter’s output at the plate. It encompasses all of the desirable factors in an advanced statistic: batting outcomes are weighted appropriately, league average is taken into account, the metric is scaled to that league average, and park and league factors are taken into account. This combination allows for the objective measurement of a batter’s output. From here, comparison to other players, disregarding any possible differences due to time period, park, etc. becomes possible. The formula for wRC+ looks like this:

$\textrm{wRC+} = \frac{(\textrm{wRAA/PA + R/PA}_{\textrm{lg}}) + (\textrm{R/PA}_{\textrm{lg}} - \textrm{PF} \cdot \textrm{R/PA}_{\textrm{lg}})}{\textrm{wRC/PA}_{\textrm{lg}}} \cdot 100$

This looks more intimidating than it actually is to compute! The equation consists of quite a few things that have already been detailed above. The first term, wRAA/PA is the same as the first half of the equation for wRAA. $\textrm{R/PA}_{\textrm{lg}}$ is the year-to-year constant used for the calculation of wRC found here, Park Factor (PF) is another constant used already, found here. The bottom term in the overall denominator, wRC/PA, is the first new term in the equation for wRC+. To calculate it, use the numbers on Fangraphs for the AL and the NL, respectively. Pull the total wRC from the column on the right, and the total number of plate appearances on the left and divide accordingly. From here, multiply by 100 (thus making league average wRC+ equal to 100).

As for OPS+, each point above and below 100 will represent 1 percentage point above or below the average wRC+. In terms of interpretation, another handy chart courtesy of Fangraphs:Note 4

Interpretation wRC wRC+
Excellent  105  160
Great  90  140
Above Average  75  115
Average  65  100
Below Average  60  80
Poor  50  75
Awful  40  60

To see how this calculation works in practicality, we will use Mike Trout’s stats one final time. In 2015, Trout had a wRC+ of:

$\textrm{wRC+} = \frac{(55.4/682 + .113) + (.113 - .94 \cdot .113)}{10594/91485} \cdot 100 = 173$

Again, there is a slight rounding error here, as Fangraphs displays 172, but the calculation is straight-forward enough. wRC+ can be more efficiently calculated in a spreadsheet, particularly if working in one throughout this page.

## Rounding Second: Sabermetric Pitching Stats and Player Valuation

This article in conjunction with the Basic Hitter Statistics provides a thorough basis of the statistics used to measure pitchers today. With this and an understanding of the Advanced Pitcher Statistics, participation in the sabermetric community is fully possible, especially here at SaberBallBlog. The knew-found knowledge can be used to engage on our discussions of players and teams. To elevate your knowledge even further, read up on WAR and Player Evaluation.  This will explain the determination of a player’s all-around ability and his worth to a team, particularly when it comes time to negotiate a contract.

1 OPS chart and explanation from Fangraphs here.
2 An on-base point is worth about 1.7 to 1.8 points of slugging, as seen here.
3 Explanation of wOBA scale.
4 Chart can be found here.