A player comes up to the plate. He’s a very good hitter; he’s hitting .300 on the year and has 40 home runs. On the mound stands a pitcher, also very good. The pitcher is a Cy Young candidate, and his ERA sits barely over 2.00. He leads the league in strikeouts and issues very few walks.

After a 10-pitch battle, the pitcher is the one to crack and the batter slaps a hanging curveball into the gap for a double. The batter has won. His batting average for the at bat is a very nice 1.000. Same for his OBP. His slugging percentage? 2.000. Fantastic. If he did this every time, he’d be MVP, no question, every year. The pitcher, meanwhile, has a WHIP for the at bat of #DIV/0!. Hasn’t even recorded a single out. His ERA is the same. He’s not doing too great. But let’s be fair. We’ll give him the benefit of the doubt, since we know he’s a good pitcher – we’ll pretend he recorded one out before this happened. Now his WHIP is 3.000. Yeesh – ugly. If he keeps pitching like this, his ERA will climb, too, since double after double after double is sure to drive every previous runner home.

Now, obviously, this is a bit ridiculous. Not every at bat is the same. The hitter won’t double every single at bat, and the pitcher won’t allow a double every time either. Baseball is a game of random variation, skill, luck, quality of opponents and teammates, and a whole bunch of other elements. In our scenario, all those elements came together to result in a two-bagger. But, like we said, you can’t expect that to happen every single time just because it happens once.

So… how do we predict what will happen in an at bat? Any person well-versed in baseball research knows that past performance against a specific batter or pitcher means little in terms of how the next at bat will turn out, at least not until you get a meaningful number of plate appearances – and even then it’s not the best tool.

Of course, if we knew the result of every at bat before it happened, it would take most of the fun out of watching. But we’re never going to be able to do that, and so we might as well try to predict as best we can. And so I have come up with a methodology for doing so that I think is very accurate and reliable, and this post is meant to present it to you.

To claim full credit for the inspiration behind this idea would be wrong; FanGraphs author and baseball-statistics aficionado Steve Staude wrote an article back in June 2013 aiming to predict the probability of a strikeout given both the batter’s and the pitcher’s strikeout rates, which led me to this topic. In that article he found a very consistent (and good) model that predicted strikeouts:

Expected Matchup K% = B x P / (0.84 x B x P + 0.16)
Where B = the batter’s historical K% against the handedness of the pitcher; and P = the pitcher’s historical K% against the handedness of the batter

He then followed that up with another article that provided an interactive tool that you could play around with to get the expected K% for a matchup of your choosing and introduced a few new formulas (mostly suggested in the comments of his first article) to provide different perspectives. It’s all very interesting stuff.

But all that gets us is K%. Which, you know, is great, and strikeouts are probably one of the most important and indicative raw numbers to know for a matchup. But that doesn’t tell us about any other stats. So as a means of following up on what he’s done (something he mentioned in the article but I have not seen any evidence of) and also as a way to find the probability of each outcome for every type of matchup (a daunting task), I did my own research.

My methodology was very similar. I took all players and plate appearances from 2003-2013 (Steve’s dataset was 2002-2012; also, I got the data all from retrosheet.org via baseballheatmaps.com – both truly indispensable resources) and for each player found their K%, BB%, 1B%, 2B%, 3B%, HR%, HBP%, and BABIP during that time. This means that a player like, say, Derek Jeter will only have his 2003-2013 stats included, not any from before 2003. I further refined that by separating each player’s numbers into vs. righty and vs. lefty numbers (Steve, in another article, proved that handedness matchups were important). I did this for both batters and pitchers. Then, for each statistic, I grouped the numbers for the batters and the numbers for the pitchers, and found the percentage of plate appearances involving a batter and a pitcher with the two grouped numbers that ended in the result in question. That’s kind of a mouthful, so let me provide an example:

These are my results for strikeout percentage (numbers here are expressed as decimals out of 1, not percentages out of 100). Total means the total proportion of plate appearances with those parameters that ended in a strikeout, while batter and pitcher mean the K% of the batter and pitcher, respectively. Count(*) measures exactly how many instances of that exact matchup there were in my data. Another important point to note – this is by no means all of the combinations that exist; in fact, for strikeouts, there were over 2,000, far more than the 20 shown here. I did have to remove many of those since there were too few observations to make meaningful assumptions…

…but I was still left with a good amount of data to work with (strikeout percentage gave me just over 400 groupings, which was plenty for my task). I went through this process for each of the rate stats that I laid out above.

My next step was to come up with a model that fit these data – in other words, estimate the total K% from the batter and pitcher K%. I did this by running a multiple regression in R, but I encountered some problems with the linearity of the data. For example, here are the results of my regression for BB% plotted against the real values for BB%:

It looks pretty good – and the r^2 of the regression line was .9653, which is excellent – but it appears to be a little bit curved. To counter that I ran a regression with the dependent variable being the natural logarithm of the total BB%, and the independent variables being the natural logarithms of the batter’s and pitcher’s BB%. After running the regression, here is what I got:

The scatterplot is much more linear, and the r^2 increased to .988. This means that ln(total) = ln(bat)*coefficient + ln(pitch)*coefficient + intercept. So if we raise both sides from the e, we get total = e^(ln(bat)*coefficient + ln(pit)*coefficient + intercept). This formula, obviously with different coefficients and intercepts, fits each of K%, BB%, 1B%, 3B%, HR%, and HBP% remarkably well; for some reason, both 2B% and BABIP did not need to be “linearized” like this and were fitted better by a simple regression without any logarithm doctoring.

Here are the regression equations, along with the r^2, for each of the stats:

The first thing that should jump out to you (or at least one of the first) is the extremely high correlation for BABIP. It totally blew my mind to think that you can find the probability, with 96% accuracy, that a batted ball will fall for a hit, given the batter’s BABIP and pitcher’s BABIP.

Another immediate observation: K%, BB%, and HBP% generally have higher correlations than 1B%, 2B%, 3B%, and HR%. This is likely due to the increased luck and randomness that a batted ball is subjected to; for example, a triple needs to have two things happen to become a triple (being put in play and falling in an area where the batter will get exactly three bases), whereas a strikeout only needs one thing to happen – the batter needs to strike out. Overall, I was very satisfied with these results, since the correlations were overall higher than I expected.

Now comes the good part – putting it all together. We have all the inputs we need to calculate many commonly-used batting stats: AVG, OBP, SLG, OPS, and wOBA. So once we input the batter and pitcher numbers, we should be able to calculate those stats with high accuracy. I developed a tool to do just that:

For a full explanation of the tool and how to use it, head over to to my (new and shiny!) blog. I encourage you to go play around with this to see the different results.

One last thing: it is important to note that I made one big assumption in doing this research that isn’t exactly true and may throw the results off a little bit. The regressions I ran were based off of results for players over their whole career (or at least the part between 2003-2013), which isn’t a great reflector of true talent level. In the long run, I think the results still will hold because there were so many data points, but in using the interactive spreadsheet, your inputs should be whatever you think is the correct reflection of a player’s true talent level (which is why I would suggest using projection systems; I think those are the best determinations of talent), and that will almost certainly not be career numbers.

Coming into this season, the Boston Red Sox had high hopes. Obviously, they were coming off a World Series title, and they had every reason to expect that they could contend again. Jarrod Saltalamacchia was gone, but he could be replaced by A.J. Pierzynski; the drop-off there wouldn’t be too large. Ryan Dempster was gone, but the Red Sox’s rotation of Jon Lester, John Lackey, Clay Buchholz, Jake Peavy, and Felix Doubront was what they had gone with during last year’s stretch run anyways. Jacoby Ellsbury was gone, but Jackie Bradley Jr. (and Grady Sizemore!) should have been able to play well enough to make his departure bearable. And Stephen Drew was gone, but uber-prospect Xander Bogaerts was ready to take over the Red Sox’s shortstop position and dominate the league.

Needless to say, none of those really worked out like the Red Sox and their fans had hoped or planned. Boston currently resides in the AL East cellar, all but certain to go from first to worst just the year after they had done the very opposite. And perhaps no individual part of that failure this season has been a bigger disappointment than Bogaerts. Instead of being the hitter he was supposed to be, he has struggled mightily at the plate, to the tune of a .223/.293/.333 slash line — good for a 74 wRC+ (as of September 1) and a major contributor to his negative WAR.

Where do we start in trying to assess the reasons for Bogaerts’s struggles? Well, time-wise, we can place a pretty neat cutoff point at June 4: That is when Bogaerts started to slump (I think I cursed him). For the first two months of the season, actually, Xander was quite good: he had a 140 wRC+ through April and May, and that figure would have been higher if not for a mini-slump that came towards the very beginning of the season. He was drawing walks roughly 11% of the time (above average) and striking out at a clip a shade below 22% (not much below average). And then came June. It started out OK — he went 4-for-13 in his first 3 June games. But after that, for the rest of the month, he recorded a mere 9 hits and 3 walks in 88 plate appearances. July was better, but not good: Bogaerts managed just a .228/.253/.342 line, and now through most of August he has been even worse than he was the previous two months, with a paltry .123/.195/.164 triple slash.

His wRC+, by month:

Yeesh. Not the way you want to be trending. So what happened? Well, the easy answer is to point to BABIP:

This looks right, right? His best month by wRC+ was his best month by BABIP. His worst month by wRC+ was his worst month by BABIP. And the same can be said for every month in between. But that, of course, doesn’t tell the whole story. Why is his BABIP from the first two months so much higher? What can he do to fix it? Will he fix it? Can he? Let’s explore.

A .364 BABIP like Bogaerts had in April is unsustainable. The .421 BABIP he had the following month is way too high for even the best players to keep up. So naturally, we would expect some regression from him. But his batted ball profile did suggest a decent BABIP – high line drive rate and low popup rate. The only thing overly suspect was his 17.1% infield hit rate in June. Nothing there would suggest such an outrageously high BABIP for the first two months, but nothing would suggest the low BABIPs that were to come later either. So something must have changed. What was it?

It wasn’t Bogaerts’s average flyball distance; that stayed more or less intact. But he did start hitting many fewer line drives…

…and started striking out more, which didn’t affect his BABIP directly but did have an impact on his overall hitting (somewhat astonishingly and coincidentally, his K% has been the exact same – to one decimal – each of the past 3 months):

And in the same vein, he walked much less, which helped contribute to his very low wRC+ as well:

So while it may be easy to ascribe Bogaerts’s recent struggles to his abnormally low BABIPs, there is more to the story. He simply isn’t hitting anywhere near as well as he did earlier in the season. I can think of a few potential reasons for this:

1. Pitchers are pitching to him differently, and he will have to adjust

2. He is in a prolonged slump, and will snap out of it eventually

3. He isn’t actually that good, and his first few months were just very lucky

4. He was playing third base

I think we can ignore the last two. Bogaerts, after all, was ranked a top-5 prospect coming into the season by almost anyone worth listening to, and he has hit very well in the majors before; he’s almost certainly not actually bad at hitting. As for the last one — that was a theory many people floated out when Bogaerts stopped hitting well at almost the exact same time as Stephen Drew returned and kicked Bogaerts over to third. The argument was that since short was Bogaerts’s natural position, and he felt most comfortable there and could focus on his hitting, he would do better when playing there.

And that theory holds some water: this season, his wRC+ as a third baseman is 37 (in 180 PA), and as a shortstop it is 95 (in 312). That is too large of a difference to dismiss offhandedly. But here’s the problem: when Drew was traded, and Bogaerts returned to shortstop, he continued to hit poorly. In fact, throughout the entire month of August, Bogaerts played shortstop, and he had a -3 wRC+. I am going to say that that theory, while compelling, doesn’t really explain Bogaerts’s struggles at all. He’d tell you that himself.

So what does? Pitchers pitching him differently? Yes, to an extent. Here is how Bogaerts has done all season long against certain pitches:

And here is how he has been pitched:

The pitches in that gif are ordered by how many runs above average Bogaerts has been against them, descending. You can see that from June 4 (the date of the start of Bogaerts’s extended slump) on, he has seen many fewer fastballs and many more sinkers and sliders than before. That could be the cause of his BABIP, strikeout, and general hitting struggles since he excels against fastballs and cannot hit sliders or sinkers (sliders more so).

But there’s only one issue: the problem isn’t that Bogaerts is getting fewer pitches he can hit, it’s that he’s not hitting the pitches he used to. Here’s Bogaerts against four-seam fastballs (from Brooks Baseball; BIP means balls in play):

He’s cut down a bit on his swings and misses, but everything else looks bad. He’s drastically decreased his line drive rate and drastically increased his popup rate. His groundball rate has gone down a bit, which can be good or bad (in this case I don’t think it’s had a huge effect on anything), and his flyball rate has gone up a lot — which could be good, but Bogaerts is averaging a mere 266.75 feet on his fly balls — 230th out of 284 qualified hitters. So how has this changed his results? Again, Bogaerts against fastballs:

Wow. That is quite the drop in production. League average wOBA against four-seamers this year is .416 (which makes you question why they are thrown so much, but that’s a different article) and so Bogaerts’s wRC+ relative to other fastballs went from a 113 to a 61 in those two timeframes (park-unadjusted).

And look where Bogaerts is hitting balls, too. The following charts aren’t only fastballs — it’s all balls put in play by him. In the beginning of the year, he was sending line drives to all fields, getting grounders through the infield, and pulling balls deep. In the second part, you see lots of shallow line drives and fly balls — in fact, in the three months covered in the second half of the gif, there are all of TWO ground balls that make it through the infield, and only one opposite-field line drive that makes it to the outfield. There are more popups, too, and the fly balls seem to be shallower generally.

Now, some of the things you’re seeing here could be a result of teams shifting on him more as the year goes on, which is why no ground balls are getting to the outfield. But more likely it is Bogaerts making weaker contact and allowing fielders to get to his ground balls; in addition, he isn’t hitting many ground balls up the middle, where you’re more likely to get hits.

Take a look at the gif above. What you’re seeing is the same thing as the last one, only with the at bat result instead of the batted ball type. In the first part of the year, you see Bogaerts getting lots of hits to all parts of the outfields, including deep balls that end up in home runs or doubles. Then, many more balls end up in the infield and most of his hits are shallow balls to the outfield.

This doesn’t look good, especially since it’s been going on for so long. I’m no expert in swing mechanics, so I can’t tell you why Bogaerts has suddenly stopped hitting everything, fastballs especially. My guess is that it’s just a long, long slump that is happening because he’s only 21 years old. I don’t think this means that we should give up on him. He has already proven that he can hit, albeit in a very small sample.

Take a look at the list of all the players who had a wRC+ below 100 in a year where they were listed as top-10 prospects by Baseball America (since 1997):

There are a lot of really good players on that list. Bogaerts is one of the worst there in terms of wRC+ that year, but he’s also younger and higher-ranked than most. That doesn’t concern me. What concerns me is that almost all of the ones on that list from the past few years haven’t succeeded: all of the ones that have are from 2009 or earlier. This is consistent with semi-recent findings by Jeff Zimmerman that the aging curve is changing: hitters don’t improve with age anymore. Further research by Brian Henry shows that players who start in the big leagues at 21 tend to stay steady with their production for a while, then decline at around 30. This does not bode well for the young Red Sox shortstop.

But who knows? If I had to guess, I would say that Bogaerts regains his stroke and starts driving the ball more. He’s too good of a hitter to be so bad against fastballs. After all, he is only 21 years old. Plus… I mean, look at that swing. Number two prospects go far. All the prospects on the list above ranked first or second had some degree of success in the majors, with the exception of Rocco Baldelli, who was good until injuries ruined his career. (Brandon Wood didn’t have enough plate appearances to qualify for the list.) If he was playing a little over his head in April and May, he’s been playing well below his feet for the past three months, and those kinds of things tend to right themselves in time.

Note: This was written before Bogaerts played today, Monday 9/1. He went 1 for 4 with a double and two strikeouts.

The plate discipline stats at FanGraphs are fantastic. Lots of stuff can be drawn from them – and the articles I’ve linked to are only scratching the surface both of what’s already been done and what we can still do with them. So many things are great about them: they’re very stable, they’re good indicators of other statistics that might be less stable, and they’re  completely isolated to the batter and pitcher. The problem is, they only go back to 2002 (for the BIS ones) or 2007 (for the Pitchf/x ones). So what if we want plate discipline numbers for players from before then? How do we know how often Babe Ruth or Willy Mays or Hank Aaron swung at pitches inside the zone, or how often they made contact on pitches outside the zone?

Regressions, that’s how.

Using the Baseball Info Solutions plate discipline data (only because it goes back farther, and also has the SwStr% and F-Strike% stats), I ran a multivariate regression with R to find all the plate discipline numbers provided on FanGraphs: O-Swing%, Z-Swing%, Swing%, O-Contact%, Z-Contact%, Contact%, Zone%, F-Strike%, and SwStr%. I used the following stats as variables in the regression: BB% and K% (for obvious reasons), ISO (I figured maybe power hitters were more prone to different types of numbers), BABIP (same goes for hitters who could maintain higher BABIPs), HR% (same thinking as ISO), and OBP (combining hitting ability and plate discipline, even if somewhat crudely). My dataset was every qualified hitting season from 2002 until now. I couldn’t use any batted ball data (GB%, FB%, etc.) as a variable because we don’t have that prior to 2002 either. So that was what I had.

Some stats worked better than others – for example, the r^2 for Contact% was an excellent 0.8089, while for Zone% it was a measly 0.1551. And of course, it’s possible that the coefficients would be different for prior eras than they are now. But, hey, what can you do. Here, first, are the r^2s for each statistic, so you know how much to trust each number:

And now for the actual coefficients:

(If you can’t see the whole table, here)

Note that for all the percentages – including the plate discipline numbers – I turned them into decimals: for example,  a BB% of 12.5% will be turned into 0.125, and  an O-Swing% of 20.7 will be 0.207, so if you’re calculating these on your own, keep that in mind.

There are some strange things in that table that I wouldn’t really expect. Here’s one: a higher O-Contact% leads to a much lower OBP, or maybe vice-versa*. The only logical explanation that I can offer is that balls out of the zone that are hit fall for hits less often, so BABIP and therefore OBP will each be lower. League average BABIP on balls out of the zone in 2013 (based on a quick search I did at Baseball Savant) was .243, well below the league average of .297. But that -1.89 coefficient still seems like too much. Some more explainable ones: HR% and Zone% are strongly inversely correlated (the more dangerous a hitter’s power, the fewer pitches they’ll see in the zone), BB% and O-Swing% are strongly inversely correlated (the fewer pitches you swing out of the zone, the more you’ll walk), and K% and SwStr% are fairly strongly correlated (the more you swing and miss, the more you’ll strike out).

To first examine these stats a little bit more, let’s take a look at the regressed numbers for players who have played since 2002 and compare them to their real numbers. Here’s Barry Bonds’s 2002 (the asterisk means it is the regressed, not real, numbers)

Hmmm… not off to the greatest start. Z-Contact, Zone, F-Strike, and Contact percentages were pretty good, but the rest were waaaay off. O-Swing gave out a negative number. As good as Barry Bonds might have been, that just isn’t possible. SwStr% is also pretty off – only pure contact hitter Marco Scutaro has ever posted a swinging strike percentage that low since the BIS data started being recorded, and nobody has every been lower. (Scutaro had 1.5% in 2013). Not terrible, though. How about Miguel Cabrera’s 2013 MVP season?

Hey, not bad! The O-Swing is pretty off, and the O-Contact is a little too low, but other than that they’re all fairly close to the real values. I think we’re getting somewhere here.

Now let’s look at some seasons for which we don’t have the real numbers. Ever wondered how Babe Ruth’s plate discipline was in 1927?

Not bad. We obviously can’t verify this (at least not without a lot of painstaking effort, and likely not at all) but that seems reasonable enough. Average contact rates in the zone, good swinging strike percentage, not very many swings outside the zone. How about the king of plate discipline, Ted Williams? Here are his numbers from his 1957 season, in which he had a 223 wRC+ and nearly 10 WAR:

Wow. Really, really good. That’s a crazy low O-Swing% and yet a fairly middle-of-the-pack Swing% overall, which goes exactly with what we would expect from a man with a famed, disciplined plate approach. He rarely swung and missed, making contact on nine out of ten swings and only whiffing on one out of every twenty five pitches he saw.

I could really go on and on, but I think I’ll end by showing you the (supposed) single worst season by these regressed plate discipline numbers between 1903 and 2001. See if you can guess who it is:

This will shock you, I’m sure, but… It’s Dave Kingman.

* Most likely, high O-Contact% causes low OBP and not vice-versa. This brings us into dangerous territory, however, because we don’t want to assume that everyone with low OBP has high O-Contact%. There are other factors that go into low OBP as well, and somebody could very easily have a low O-Contact% and a low OBP. It is like this with each of the regressed stats. But this is the best I could really do.

Pitching statistics are mostly based on rates. Sure, we have innings pitched, and if you want to annoy me, you can talk about wins and losses, and of course there are the “three true outcomes” of strikeouts, walks, and home runs, plus WAR. But nobody ever looks at how many runs a pitcher was above or below average. Runs allowed isn’t all that common of a statistic; you’re more likely to see ERA or RA9. Even strikeouts and walks are often expressed as a percentage of all plate appearances, or as an amount per nine innings. The defense-independent ERA estimators like FIP and its spinoffs are rates, just like ERA. Where batters have regressed plus-minus or counting stats like wRAA and wRC, pitchers have nothing.

However, there has got to be some value in counting stats for pitchers. If we want to know how many more or fewer runs a team would allow by putting in an average pitcher instead of any given pitcher, that statistic would be able to tell us. So I’m going to present here three basic different numbers, one based off of FIP, one based off of straight runs allowed, and the third based off of linear weights. Each will be in two forms – raw runs allowed and runs allowed above or below average. I’ll call them FIP-Runs and FIP-Runs Above Average (FIPRAA), wRC-Runs and wRC-Runs Above Average (wRCRAA), and, obviously Runs and Runs Above Average (RAA). Kind of long, yeah, but I didn’t want to call the FIP one FRAA because that already exists.

All data was obtained from FanGraphs except for the singles, doubles, triples, home runs, walks, and HBP against used to calculate wRC; FanGraphs does not have some of those so I used Baseball-Reference.

FIP-Runs

This should be pretty simple. Take a pitcher’s FIP. FIP is scaled like ERA, but we want to scale it to RA9 because we want to scale it to all the runs a pitcher allows, not just the earned runs. To do this, multiply it by a constant that changes yearly – for 2013, it was 1.08. This is the league RA9 divided by the league ERA.

Take that figure, multiply it by the number of innings they pitched, and divide by nine to get the number of runs that FIP says a pitcher should have allowed. That’s their FIP-Runs. Great. But now how do we get that to express how many runs above average they were worth?

Well, we already have FIP-, which tells us how much better a pitcher’s FIP was than league average – and it’s already park- and league-adjusted, to boot. So what I did was subtract each pitcher’s FIP- from 200 to get the inverse of their FIP- (so if a pitcher had a 90 FIP-, the inverse would be 110) and multiplied that by their FIP-Runs. That gave me the number of runs (adjusted to the park and league) that an average pitcher would give up in the same number of innings. Just subtract the pitcher’s FIP-Runs, and you have your FIPRAA. And while I was at it, I did the same thing for xFIP. You can find the numbers at the end of the article. But first, the next part:

Runs

I didn’t have to calculate these like I did with FIP-Runs because the numbers are already there – it’s just the total number of runs a pitcher allowed. I did, however, have to calculate RAA, for which I used the same method as I did with FIP-Runs: find the RA9- (this was not park- or league-adjusted because I calculated it myself), take the inverse, multiply it by the runs, and subtract the runs from that. Piece of cake. Now for the last, and hardest, part of this:

wRC-Runs

These were tricky. I had to calculate each pitcher’s wRC against by first finding their wOBA against with the raw number of singles, doubles, triples, etc. they gave up and converting that into wRC. (I’ve actually already put this in a community post in a different form). But from there, I could follow the same instructions as before: use the wRC against as runs allowed, find the wRC/9- (if you didn’t read the article I linked to earlier, wRC/9 is just wRC against scaled like RA/9), and from those two find the wRCRAA (quite a mouthful, I know).

So, without further ado, here are the numbers (sorted by FIPRAA):