Better Match-Up Data: Forecasting Strikeout Rate

“Riddle me this,” wrote editor Dave Cameron to me some time ago, “what happens when an unstoppable force meets an immovable object?”  OK, that’s not exactly how it went down.  What he actually did was to present me with the challenge of research, with the goal being to develop a model that would forecast the expected odds of an outcome of each match-up between a specific batter and a specific pitcher. Rather than talking about how players have done in small samples, can we use our understanding of player skillsets to develop an expected outcome matrix for each at-bat?

For example, such a tool might tell you that Adam Dunn has a 40% chance of striking out against Stephen Strasburg, a 10% chance of drawing a walk, a 5% chance of hitting a ground ball, etc… Forget I said those particular numbers — I completely made them up in my head just now.  You may be thinking “well, why should I care about that?  Rather than just being inundated with match-up data that is little more than randomness, such a tool might give you some idea of how much of a gain in expected strikeout rate a team would get by switching relief pitchers with a man on third base and less than two out. Or what the probability of getting a ground ball is in a double play situation, which might influence the decision of whether or not to bunt. Knowing the odds of potential outcomes could be quite beneficial in understanding the risks and rewards of various in-game decisions.

This project has been — and will continue to be — a major undertaking, as you can imagine.  This isn’t the kind of thing that can just be thrown together, but I really think the results could be great. Today, I’ll be sharing with you the findings of my research into perhaps the most important aspect of these matchups — K%, or strikeouts per plate appearance.  This will introduce the sort of process that will be involved in figuring out all of the other elements of the matchup tool.

Now, I think it’s common sense that when you match up a batter who strikes out a lot against a pitcher who whiffs a lot of batters, the batter will probably strike out at a very high rate.  Conversely, a batter who rarely strikes out will likely strike out even less against a “pitch to contact” hurler.  But what happens when a high-K batter faces an equally low-K pitcher?  The unstoppably-swinging bat against the barely-moving pitch, you might say.  Is the resulting K% average, above average, or below average?  Does the batter or the pitcher exert more influence on the results?  And can the results even be predicted?  There’s only one way to be sure — look at a ton of data.

And a ton of data I did look at — over 1.5 million plate appearances, all told.  Alright, my computer did most of the looking for me, to give credit where it’s due.   The data set was play-by-play data from 2002 through 2012.  Originally, there were close to 2 million PAs, but many were thrown out because they involved a non-qualifying pitcher or batter.

What made a player “qualify” for consideration here?  Well, first of all, the analysis does take into account the handedness of both the batter and pitcher in each PA.  If a particular PA involved a left-handed batter (LHB) against a right-handed pitcher (RHP), then what I did was to analyze the results of the PA through the lenses of the batter’s historical K% against RHP and the pitcher’s historical K% against LHB.  By “historical,” I mean their average over the entirety of 2002-2012.

Now, I’ll understand if some of you take issue with that, because players don’t necessarily stay the same player for [up to] 11 seasons, but hopefully you’ll understand what I mean when I say that a big part of why I did that was to keep things from getting too incestuous.  There’s some circularity going on here, when you think about it, as the PA makes up part of the history I’m drawing from; better to make that PA a very small part of the history, I say.

Anyway, to qualify for consideration, a pitcher needed at least 600 PA against the handedness of the batter in question, and a batter needed at least 300 PA against the handedness of pitcher in question.  Admittedly, they’re mainly arbitrary cutoffs, but the reasoning behind them has to do with stability, as well as the rarity of LHP relative to LHB (especially when you include switch-hitters).

Since looking at every possible combination of K% down to two decimal places (e.g. 7.64% K batters vs. 21.39% K pitchers) would make things very unwieldy, what I did was to round every player’s historical numbers off to the nearest percent.  I then looked at all the different pitcher-batter combinations in range, now that they were manageable.  I found, for example, 9,536 plate appearances that involved a batter with around a 15% K rate and a pitcher with an approximate 13% K rate.  Out of these PAs, 998 of them resulted in a K, which means the observed K% for a 15% vs. 13% matchup was about 10.5%.

“But wait,” you might be thinking, “why would the resulting K% be lower than either of the individual rates?”  Well, for one thing, the league average K% over the period was 17.5%.  That means that the batters in this scenario have established their 15% K rates against pitchers who are, on average, better at striking batters out than these 13% K pitchers.  It also means that these 13% K pitchers are used to seeing batters who are a little bit easier to strike out.  The net result is that this group of pitchers has a harder time, while the hitters have an easier time than usual, hence the particularly low resulting K% for this matchup.

Here’s a chart to give you an idea of how well-represented the different K% levels were in the sample:

As you can see, it’s not quite a bell curve (a.k.a. normal distribution), but is skewed a bit, such that there are more PAs to the right of the peaks than to the left (“positive skew”).  That’s to be expected, seeing as how 0% is as low as you can possibly go.

Results

Now, here’s a little sample of some of the K% rates I found for the different matchups:

 Pitcher’s K% 10% 15% 20% 25% Batter’s K% 10% 5.3% 7.8% 12.0% 14.6% 15% 8.1% 11.4% 16.0% 20.8% 20% 11.8% 15.6% 20.6% 26.9% 25% 13.6% 21.0% 24.6% 34.9%

That 25-25 matchup result is a bit of an outlier, by the way (31% would be more in-line with what surrounds it).  It’s based on only 759 PAs, compared with 10,704 PAs for the 15-15 matchup, for example, though.  There just are fewer batters and pitchers who strike out or get struck out 25% of the time.  The results towards the extremes are therefore going to be less reliable.  That’s why for the next chart, I only considered matchups with at least 1200 PAs to represent them.  309 matchup types made that cut, by the way.

So, it might take you a few seconds to understand what you’re seeing in this chart, if you’ve never seen one of these before.  Just imagine you’re seeing a 3D object, with the “8%” you see at the very bottom being at the nearest corner to you of the “box” that contains the shape.  The height represents the observed K% for each matchup, and the different color slices each representing a 5% interval of the observed K%.  The graph treats non-existent data points as having zero height, so don’t make anything of the blue “floor” there other than that there weren’t enough plate appearances for those types of matchups to make the cut here.  Here you go:

It’s not perfectly smooth, probably mainly due to random variation, but there’s a pretty clear trend here.  It’s forming basically a flat surface on top.  Hopefully this isn’t too confusing, but if you were to tilt the box that contains this shape so that this side edge was pointing straight at you, it would look like this:

So, that’s looking at it so that the Batter’s K% axis and Pitcher’s K% axis are both angled 45 degrees from facing directly towards you, which mean you’re seeing a balanced view of the two of them.  And what do we see?  It’s pretty danged close to a straight line that the top surface is making here.  It bulges out a bit towards the top (in the high K% areas), but that’s to be expected due to the relatively fewer PAs being represented there.  The peaks are higher, but the valleys are lower in the high K% areas due to more randomness, is what I’m saying.

Just for fun, here’s what happens to the chart before the last one when the PA requirement for inclusion for a point is lowered to 100 PA (there are now 634 unique matchup combinations represented here):

It looks more random, as expected, but I’d say the pattern still holds up very well.

What Does this All Mean?

To me, the findings suggest that there is a very strong trend at work here, and probably a good amount of predictability, at least in the overall sense.  When it comes to particular players, sure, there might be some batters with a particular weakness against a filthy curve, or a blazing fastball.  Making those sorts of adjustments might come in a later incarnation of the matchup tool, but for now, I think this gives us a pretty solid baseline to work off of.

As for whether the pitcher or batter exerts more influence on the outcome, I’d have to say, in the overall sense, it’s pretty much a tie.  Maybe eyeballing a graph of the data isn’t enough proof of that for you.  Well, if you know my work, you had to know I’d give you a regression formula or two.  As it turns out, there’s a very simple formula I derived that explains a remarkable amount of what we see here in the data (besides the randomness, of course), and it will help to prove that point:

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

As you can see, the batter and pitcher contribute equally to the outcome in this formula.

In the next installment, I’ll show you that the formula (and a slightly more fine-tuned variation of it) does a pretty good job of explaining the trends in the data.  One of these formulas (or something like them) will probably be the basis of the K% aspect of the matchup tool.  I’ll also provide you with a better look at the data, for those interested, as well as an expected matchup K% calculator.  We’ll even see how well we can predict the results of 2013 matchups based on past data.

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46 Responses to “Better Match-Up Data: Forecasting Strikeout Rate”

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1. Neil says:

I’ve often thought about matchups this way (type of hitter vs type of pitcher), so I’m happy to see someone taking this project on. My one complaint is that I would have liked to see the ratios built from 2002-2011 and tested on 2012. I know you plan to do it with 2013, but it’s a little better practice to show something like this working when you present it.

That’s probably the academic in me talking, but I love where this is going.

• Thanks everybody!

Neil — yeah, I’m also wishing I’d done that, but the 2013 results look good regardless. And you’ll probably be seeing those in a day or two.

2. Anon says:

Fangraphs… Now in 3D!

Awesome graphs and awesome work.

I’ve been eagerly waiting for more research and statistics that include opponent quality for quite a while. Thanks, and keep it up.

3. johnnycuff says:

exciting stuff. i’m very much looking forward to the rest of this series.

4. Dan says:

This is the coolest research I’ve seen on Fangraphs

5. Justin says:

This is fantastic.

6. Ned Yost says:

All those words and geometric doo-dads were fun to stare at for the last hour.

But it still doesn’t beat thirty years of baseball.

• Stats803 says:

So what you’re saying is that 30 years of experience watching baseball can predict a batter/pitcher matchup’s expected K% better than this data?

• Ned Yost says:

It’s a joke. Ned Yost, manager of the Kansas City Royals, predicted in the first few days of spring training this year that Luke Hochever would win 18 games and potentially the Cy Young Award. (He’s doing okay out of the pen and hasn’t started a game this year) Reporter asked him what he’s basing that on, and he replied, “30 years of baseball.” It’s sort of a running joke in Kansas City. It has pretty wide application even in non-baseball situations if you like the joke, which usually means you recognize that Ned Yost is an idiot.

• Stats803 says:

haha, well damn, I didn’t catch that. I guess lump me in with Yost in the idiot section, then. I hadn’t heard that quote, so I apologize for my comment. Carry on!

7. Peter Jensen says:

Steve – Before you go any further it may be helpful to look at discussions on the odds ratio method at Tango’s the Book Blog and also http://www.tangotiger.net/wiki/index.php?title=Mailbags&diff=cur&oldid=1430.

• Thanks Peter. I’m away from my spreadsheets but I’ll certainly try that out tonight.

8. Matthew M. says:

Really enjoyed reading this. Just curious — is there a reason that your regression produces a curved surface (similar to a quarter of a dome) rather than a plane since the data looks planar? It also seems to apply boundary conditions that Expected Matchup K% (0,P) = Expected Matchup K% (B,0) = 0 and Expected Matchup K% (1,1) = 1. Obviously there aren’t batters that strike out 0% of the time nor pitchers that strike batters out 0% so these will probably be extrapolated from the data, but I just wondering if you were imposing these boundary conditions or if they were a result of the regression? Thanks.

• Good point. Might need a new formula. Yes, I did set those boundaries with dummy variables.

9. Ben says:

Suggestion: The data may fit the equation given in the article, but the probability involved would suggest that the equation of that surface should really be something like:
``` c1*Kb+c2*Kp-c3*Kb*Kp ```
This fits nicely with Matthew M.’s comment above that the surface really is curved; there should be at least a small quadratic effect on the data.

I notice also that while the data is almost symmetrical across batter rate vs. pitcher rate, the pitcher rate seems to make slightly more difference. This makes sense when you think about it: at some level a pitcher with filthy stuff will be able to strike out even the best contact hitter.

• Thanks Ben. I think I tried that, but I’ll give it a shot tonight if not.

• I tried it out, Ben, but wasn’t really able to make it work. It either works OK for a while before massively overestimating the high K% areas, or it doesn’t overestimate the high K% areas, yet is off otherwise.

10. Big Sexy says:

Great article.

11. ettin says:

I like pretty pictures! Pretty pictures good….

12. MGL says:

Steve, you REALLY need to include the league K rate. Any regression formula, including yours, implies a league K rate (the average K rate from 02-12).

If the league rate were different, than the formula would be different. For example, if a batter and pitcher had a 10% K rate in a league with a 10% K rate, then obviously the batter against this pitcher would be 10% as well. But if the league rate were 15%, then obviously the batter would be facing a pitcher who K’s less than the league rate so his K rate would be less as well.

So you need to normalize one or the other (or both) of the rates before doing the analysis and getting a regression equation, or you need to add the league K rate as one of the variables.

The question, “How often will a batter who strikes out 10% of the time strike out against a pitcher who K’s his opponents 10% of the time?” is not an answerable question without knowing the league rate (i.e., the batter K’s 10% of the time in what environment?)

Now, while your formula may work because it implies a certain league rate (again, the average rate from 02-12), it is too general a formula to be used with individual batters and pitchers (without normalizing their K rates or including the league rate as another variable), unless that particular batter and pitcher happened to play in that time period (02-12) such that their K rates were against a league rate that is implied by your formula.

For example, your formula might imply a league rate of 15% (I think you said it was 17.5%, but it doesn’t matter what it actually was for the point I am making), but if you want to know how batter X will do against pitcher Y and batter X or pitcher Y played when the league rate was 20% or 10% rather than the 15% for all the years combined, then your formula will be wrong.

All that being said, are you simply re-inventing odds ratio? I was under the impression that the odds ratio formula almost perfectly predicts the outcomes you are looking for. And the odds ratio of course uses the league average rate in addition to the batter’s and pitcher’s rates. I am not sure that any analysis like you are doing or resultant regression formula is going to do better than the odds ratio, but I am not sure about that. I am sure that you have to include league rates though.

• Thanks MGL. Yeah, you’re right about league average needing to be a factor, of course (this formula is tailored only to a specific era).

Regarding the odds ratio — am I correct that under the league average of 17.5% over the span, the odds ratio calculation says that the 15%-20% matchups should produce a K 20.8% of the time? Because thatwould be a massive overestimate, according to the data, which have it around 16%.

I’m wondering if an assumption of the odds ratio calculation is a normal distribution; K% is definitely skewed.

13. jsolid says:

This problem can be solved using probability. (Forgive the text syntax that is about to happen.) Each at bat has several possible outcomes. For simplicity, lets assume K, BB, HR, Other. (This can be extended to as many outcomes as you like, the more outcomes specified the better the model.) I chose the notation Kp=pitcher probability of a K, Kb=batter probability of a K, etc. Then the probability of a strikeout is:

P(strikeout)=[Kp*Kb]/[Kp*Kb+BBp*BBh+HRp*HRh+Op*Oh]

I will try to explain the math briefly… apologies in advance. Pitcher and batter probabilities are treated as independent events. The numerator is all events which are a strikeout, which is when the pitcher and batter both have a strikeout. To find that probability, multiply the individual probabilities. The denominator is all possible events, which are K,BB,HR,other and can only occur when pitcher and batter outcomes agree. Again, multiply the pitcher and batter probabilities for matching outcomes, then add the results. This is, roughly, P(K)/[P(K)+P(BB)+P(HR)+P(Other)]. The formula is similar for any outcome. (By the way, the same approach can be used to prove the Bill James log5 win percentage formula for team matchups, where the two outcomes are Win and Loss, and outcome pairs must be different not matching.)

The assumption that batter probabilities and pitcher probabilities are independent of course in reality is not true. It might be interesting to see how the theoretical results diverge from the data, and if there are any predictable patterns of that. For example, certain types of hitter do better or worse, outliers at the extremes, etc.

• Interesting, thanks. Pretty similar to my formula, but taking a lot more factors into account. I’ll try to see if I can test it out a bit.

• MGL says:

jsolid, it is not necessary to use the P for the other events – merely K or non-K, p and q (1-p).

14. Pizza Cutter says:

I’d be interested to see how this model performs against the standard “odds ratio” model. Convert your probabilities to odds ratios (p/(1-p)) for the batter, pitcher, and league.

The formula is easy enough:

(Expected OR / League OR) = (Batter OR / League OR) * (Pitcher OR / League OR)

That simplifies to Expected OR = Batter OR * Pitcher OR / League OR.

Convert OR back to probability by OR / (OR + 1)

• Thanks Pizza, I’ll definitely look into it. Like I said to MGL, is normality an assumption of that? I’m wondering how the skewness of K% might be accounted for if so.

• J. Cross says:

So, using the formula that Pizza/MGL are suggesting:

result_odds = pitcher odds * hitter odds / league odds

you can rearrange that into:

log(result_odds) = log(pitcher odds) + log(hitter odds) – log(league odds)

which implies that you could check it by doing a logistic regression predicting results (K or not K) with 3 predictors: log(pitcher odds), log(hitter odds) and log(league odds).

If you get coefficients that are distinguishable from 1 then maybe the good old odds ratio isn’t working? If not, I’d lean towards just using the odds ratio.

• Awesome, thanks Jared. Yeah, the first thing I tried was a logistic regression, but I was using probabilities instead of odds. So I’ll give that a shot.

• Pizza Cutter says:

Steve, if you’re going to use logit, take those probabilities, convert them to odds ratios and then take the natural log (log base e) of those odds ratios. It’s a trick I use all the time. In the logit regression, the coefficient should be close to 1.00, and it should be highly significant.

• Thanks Pizza. Yeah, I figured out the natural log bit last night. I’m getting as my coefficients:
Batters: 0.914
Pitchers: 0.920
League K%: 0.771

The results are great, though. I mean, they’re about the same as my “fine-tuned” formula over the ’02-’12 data set (haven’t tried it on 2013 yet) — BP/(1.066BP – .12B -.149P +.2031) — but it does make more sense at hypothetical levels, now that I think of it.

If you’re thinking those coefficients sound too low… well, I found the plain Odds Ratio, despite having a correlation on par with those of the other formulas, to be miscalibrated over the ’02-’12 set, overestimating by more and more the higher the observed K% went.

15. MGL says:

” am I correct that under the league average of 17.5% over the span, the odds ratio calculation says that the 15%-20% matchups should produce a K 20.8% of the time?”

I think odds ratio puts that at 17.2% for the expected K rate, not 20.8%.

The simple way to do the matchup is to simply add 2.5% to the batter’s 15%, to get 17.5% which is close to the odds ratio result. The 2.5% is 20% (pitcher) minus 17.5% (league).

I’m not sure that this is not good enough. Whatever formula you come up with using empirical data, because of the sample sizes, the uncertainty around that formula is going to be large. We cannot tell whether that formula is correct (and captures some interdependence) or some theoretical one which assumes independence, like the one from jsolid, is correct.

I am not sure that jsolid’s formula is different from the odds ratio one. I think that the odds ratio formula is derived from the probabilities assuming independence, but I am not sure. At least that is true for calculating the chances of team A beating team B in a game if you know each of their respective WP.

BTW, you HAVE to us in-sample data when deriving a formula, like Steve did, or when comparing results to some formula like the odds ratio. If you use out of sample data for the results, or the independent variable (the K rate for a particular matchup), then you are going to include regression toward the mean. For example, if in a league where the K rate is 17.5% and you have pitchers with a rate of 25% facing batters with a rate of 25%, and then you look at out of sample data for those same pitchers and batters (where you don’t know the pitchers’ and batters’ rates anymore), first of all, the new rates for the pitchers and batters will be less than 25% – they will have regressed towards the league mean of 17.5%, so the resultant K rate will be a result of pitchers and batters with something like 19% K rates and not 20%…

• OK thanks. I missed the final step of the calculations (thanks Russell for providing the instructions). I’ll try it out in time for Article 2. 17.2% is much better, but could the skewness of K% could lead to overestimates?

16. tz says:

“This project has been — and will continue to be — a major undertaking…. but I really think the results could be great.”

Potentially a huge understatement. Just having statistical support that batters and pitchers have roughly the same impact on K% rates for any given PA is huge.

I can’t wait for future installments. GREAT stuff!!!!

17. MGL says:

“Potentially a huge understatement. Just having statistical support that batters and pitchers have roughly the same impact on K% rates for any given PA is huge.”

First of all, we have always assumed that if you matchup any component, that the result can be derived from the odds ratio method or something close to that.

Second of all, I’m not sure that saying that “pitchers and batters have roughly the same impact,” is the right use of words here. Typically we measure the “impact” of one player or the other in a confrontation by the spread of talent within each group independent of the other group. For example, let’s say that all batters roughly the same K rate, i.e., the variance of true talent K rate among batters was very small. Let’s say that the high K batters where 18% and the low K ones were 16%. And let’s say that among pitchers, the spread was large, say 10% to 25%. Typically we would say that the pitchers “controlled” the K outcome of a matchup far more than the batters. We can even reduce that to, “Pitchers completely control the outcome of a PA with regard to K outcome or percentage,” if all batters had the same true talent K rate.

Yet, even if batters were from 16 to 18% and pitchers from 10 to 25%, we would still use the odds ratio or something like that to determine the expected outcome of a particular batter/pitcher matchup. So again, I don’t really like the concept of them having “roughly the same impact,” but we are talking semantics here.

Did we debunk the “conventional” concept of “good pitching stops good hitting” or something like that? Well, if we ignore everything that has been discovered in the last 30 years, I guess we did.

I mean, every credible sabermetrician or baseball analyst on the planet who models games uses an odds ratio or log 5 to estimate the expected outcome of a PA. I have been doing that for over 20 years.

Not to denigrate Steve’s work here. It is good. It is very worthwhile to see see how much the actual results match up to the traditional method of using odds ratio, multiplicative, or additive methods (additive is batter plus (pitcher-lg), and multiplicative is batter times pitcher/lg)…

18. MGL says:

“17.2% is much better, but could the skewness of K% could lead to overestimates?”

I don’t know that the odds ratio method relies on the existence of a particular distribution, like the normal one. I think it only requires independence in the two probabilities. I could be wrong about that. I think these are questions that can be answered by a statistician. Maybe Pizza can chime in.

What could lead to over and underestimates, and is likely in fact, at least to some extent, is dependence. For one thing, pitchers are likely to change their style and thus their underlying true talent rates against different types of batters and vice versa.

19. Jon L. says:

The one thing that really could have improved this analysis would be if you could have found that either pitchers or hitters were much more important in determining K% – for instance, if it were 75% pitchers.

You should work on that.

20. Ecological Fallacy says:

This is a perfect example of an ecological fallacy; it would be incorrect to draw inferences about individual players from group-level data.

• Why should I trust you, Ecological Fallacy? You are, after all, a Fallacy…

Seriously, though — can you explain what hidden, underlying mechanisms could be at work here that would cause an extremely strong, yet misleading trend?

21. Jay29 says:

If you use Steve’s formula for Dunn vs Sabathia using 2010-13 K% data, you get a 45.3% expected K rate! (In that span, Dunn is actually 0 for 7 vs CC with 4 Ks — a noisy 57.1% rate.)

Using 2002-13 data as suggested yields a 37.1% predicted K-rate (actual result is 34.3%).

Just thought I’d share what seemed like an extreme example.

• Jay29 says:

Wow, the lightest K% matchup I could find for 02-13 data was Aaron Cook (7.8% vs L) vs Juan Pierre (5.9% vs R)… resulting in a 2.8% expected K rate! That’s amazing. They actually combined for a 3.3% K rate over those years (30 PA).

Heh, I’m really liking this research so far. Thanks, Steve!

• Awesome, thanks Jay! I made an interactive calculator for the next article that will make it easier for people to figure these out (for all the formulas now, including the odds ratio ones).

It will even tell you how big of a role randomness plays, based on how many PA you’re looking at —
e.g. it tells you since there are only 30 PA between Cook and Pierre, you’d expect the actual strikeout rate between them to be within 1% of 2.8% (i.e. 1.8% to 3.8%) less than 37% of the time (so it got kind of lucky there, to be that close).

And for the Dunn-Sabathia one, it says there was at best a 29% chance of the observed K% being even within 10% of 45.3%, given only 7 PA.

22. Cyril Morong says:

“I found, for example, 9,536 plate appearances that involved a batter with around a 15% K rate and a pitcher with an approximate 13% K rate. Out of these PAs, 998 of them resulted in a K, which means the observed K% for a 15% vs. 13% matchup was about 10.5%.

“But wait,” you might be thinking, “why would the resulting K% be lower than either of the individual rates?” Well, for one thing, the league average K% over the period was 17.5%. That means that the batters in this scenario have established their 15% K rates against pitchers who are, on average, better at striking batters out than these 13% K pitchers. It also means that these 13% K pitchers are used to seeing batters who are a little bit easier to strike out. The net result is that this group of pitchers has a harder time, while the hitters have an easier time than usual, hence the particularly low resulting K% for this matchup.”

What I recall about the Strat-O-Matic game is that the batter cards and pitcher cards were based on deviations from the league averge. And the game was set up so you used each card half the time.

So if the league average is 17.5% and a batter actually struck out 15% of the time, on his card he would have 12.5% strikeouts since he is 2.5% below the league average (well, 2.5 percentage points). The 15 – 2.5 = 12.5. So half the time he gets 12.5% and the other half (being all of the pitcher cards collectively) is 17.5%. So if you played a whole season, he would strikeout 15%.

Something similar for the pitcher. He would have 8.5% on his card because that is 4.5% below what he actually got which is 4.5% below the league average.

So if you were on the batter’s card half the time with 12.5% and on the pitcher’s card half the time with 8.5% you would get 10.5% overall. That is what Steve predicts.

Now SOM may not work for extreme guys. What about a pitcher who only struck out 5% of the batters. You would have to go negative. But you cannot have negative strikeouts

• Thanks, Cyril! Oh man, I would have loved to play Strat-O-Matic, if I’d had any friends who were into that sort of thing.

Allow me to plug your sabermetric research for you: http://cyrilmorong.com/ I’ll try to get caught up on all that!

23. Cyril Morong says:

Not clear if anyone has said this, but have you looked at the formula at this page?

http://www.baseballthinkfactory.org/btf/scholars/levitt/articles/batter_pitcher_matchup.htm