# Another mathematician visits the ballpark – OPS

Yes, I more-or-less stole the above title from the 2004 Ken Ross book entitled A Mathematician at the Ballpark. Like that book, anyone familiar with middle-school (or junior high school) math, should have no problem with most of what we do here. However, I will try to go into baseball in more detail than the book did.

Paraphrasing slightly, I read somewhere the following facetious remark:

From a survey of 1000 random baseball fans

across the nation,  183% of them hate math.

If you are one of these 183%, then this series could be for you. Hopefully, even if you aren’t a baseball expert, but you would like to learn some baseball statistics, (now often called “sabermetrics”), these posts will help. I’m no expert myself, so we’ll learn together.

In this series of blog posts, each post will introduce a particular metric in baseball statistics as well as some of the math and baseball behind it. We illustrate all these notions using the Baltimore Orioles’ 2022 season.

This week we look at one of the most popular statistics you see on televised games: OPS or “On-base Plus Slugging,” which is short for on-base percentage plus slugging percentage. Don’t worry, we’ll explain all these terms as we go.

### On-base percentage

First, On-Base Percentage or OBP is a more recent version of On-Base Average or OBA (the same as OBP but the SF term is omitted). We define

OBP = (H + BB + HBP)/(AB + BB + HBP + SF),

where

• H is the number of Hits (the times the batter reaches base because of a batted, fair ball without error by the defense),
• BB is the number of Base-on-Balls (or walks), where a batter receives four pitches that the umpire calls balls, and is in turn awarded first base,
• HBP, or Hit By Pitch, counts the times this hitter is touched by a pitch and awarded first base as a result, and
• SF is the number of Sacrifice Flies and AB the number of At-Bats, which are more complicated to carefully define.

The official scorer keeps track of all these numbers, and more, as the baseball game is played. We still have to define the expressions AB and SF.

First, SF, or Sacrifice Flies, counts the number of fly balls hit to the outfield for which both of the following are true:

• this fly is caught for an out, and a baserunner scores after the catch (so there must be at most one hit at the time),
• the fly is dropped, and a runner scores, if in the scorer’s judgment the runner could have scored after the catch had the fly ball been caught.

A sacrifice fly is only credited if a runner scores on the play. (By the way, this is a “recent” statistic, as they weren’t tabulated before 1954. Between 1876, when the major league baseball national league was born, and 1954 baseball analysts used the OBA instead.)

Second, AB, or At-Bats, counts those plate appearances that are not one of the following:

• A walk,
• being hit by a pitch,
• a bunt (or Sacrifice Hit, SH),
• a sacrifice fly,
• interference (the catcher hitting the bat with his glove, for example), or
• an obstruction (by the first baseman blocking the base path, for example).

Incidentally, the self-explanatory number Plate Appearances, or PA, can differ from AB by as much as 10%, mostly due to the number of walks that a batter can draw.

The main terms in the OBP expression are H and AB. So we naturally expect OBP to be approximately equal to the Batting Average, defined by

BA = H/AB,

For example, if we take the top 18 Orioles players and plot the BA vs the OBP, we get the following graph:

The line shown above is simply the line of best fit to visually indicate the correlation.

Example: As an example, let’s look at the Orioles’ All-Star center fielder,  Curtis Mullins, who had 672 plate appearances and 608 at bats, for a difference of 672 − 608 = 64. He had 1 bunt, 5 sacrifice flies, he was hit by a pitch 9 times, and walked 47 times. These add up to 62, so (using the above definition of AB) the number of times he was awarded 1st base due to interference or obstruction was 64 − 62 = 2.

Mullins’ H = 157 hits break down into 105 singles, 32 doubles, 4 triples, and 16 home runs.

Second, let’s add to these his 126 strikeouts, for a total of 157+126+64 = 347.

The remaining 608 − 347 = 261 plate appearances were pitches hit by Mullins, but either caught on the fly but a fielder or the ball landed fair and Mullins was thrown out at a base.

These account for all of Mullins’ plate appearances. Mullins has a batting average of BA = 157/608 = 0.258 and an on-base percentage of OBP = 0.318.

### Slugging percentage

The slugging percentage, SLG, (SLuGging) is the total bases achieved on hits divided by at-bats:

SLG = TB/AB.

Here, TB or Total Bases, is the weighted sum

TB = 1B + 2*2B + 3*3B + 4*HR,

where

• 1B is the number of “singles” (hits where the batter makes it to 1st Base),
• 2B is the number of doubles,
• 3B is the number of triples, and
• HR denotes the number of Home Runs.

### On-base Plus Slugging

With all these definitions under own belt, finally we are ready to compute “on-base plus slugging”, that is the on-base percentage plus slugging percentage:

OPS = OBP + SLG.

Example: Again, let’s consider Curtis Mullins. He had 1B = 105 singles, 2B = 32 doubles, 3B = 4 triples, and HR = 16 home runs, so his TB = 105+64+12+64 = 245. Therefore, his SLG = 245/608 = 0.403, so his on-base plus slugging is OPS = OBP + SLG = 0.318 + 0.403 = 0.721.

This finishes our discussion of OPS. I hope this helps explain it better. For more, see the OPS page at the MLB site or the wikipedia page for OPS