Just so people know, the timeliness is my fault - I was out of town all of last week due to a family emergency and my internet access was extremely limited.

One guy in the comments thread was a howling Ludwick fan, and I pretty much dismissed him. If you’re he and you’re reading this, can you please tell me how you knew Ludwick would be healthy? I’m poor at predicting injuries. You’re either good or you were lucky.

I just don't predict injury on the individual level. As you pointed out your pool style more or less accounts for injuries so why worry about the injuries as a whole and instead concentrate on how likely it is for a player to be good and how good. (this is for position players, chronically injured players like Drew or pitchers I may take exception, but Ludwicks injuries were really just two major injuries, nothing to put a label as chronicaly injured) And watching Ludwicks bat at the end of 2007 the ball exploded off of it, factor in his years as a prospect and that his injuries only delayed his arrival to the show, not ended it, and to me it was fairly straight forward that he can flat out hit. The only real issue was whether TLR would give him the playing time to prove it.


I like the way you explained your pool system, it makes a lot more sense than looking at just the starters and projecting out from there, especially on a TLR led team.

Doing this for the Marlins:

Bat Pool Cantu 1B
Ross RF/CF
Hermida LF
Maybin CF
Gload 1B/RF
Carroll LF/RF

Glove Pool Uggla 2B
Ramirez SS
Bonifacio 3B
Helms 3B
Amezaga INF

Catcher Pool Baker C
Paulino C

Pitcher Pool Nolasco
Johnson
Volstad
Sanchez
Miller
Kensing

Closer/Setup Lindstrom
Nunez

Relief Pool Pinto
Calero
Meyer
Penn

I like this system. Good lord does this team need the pitching staff to live up to its promise and for a couple of offensive players to suddenly announce themselves (Baker, Maybin, Ross, Bonifacio). I figure there's little chance Bonifacio or Ross will surprise, but Maybin and Baker might.

Cardsfanboy-

1. Thank you! I had no idea how this giant thing would be received.

2. You were paying attention to Ryan Ludwick's swing and how hard he hit the ball? In 2007??!!! WOW, do YOU follow baseball! Remind me to ask you for details when I'm trying to evaluate a Cards' prospect.

3. The pool format was one of the very nice things that came out of a 7-year stint writing about baseball for the local alternative weekly, the Riverfront Times. That paper had a very effective but odd sports staff, until it got sold to a chain and they cancelled sports altogether. Football was Howard Balzer, who only had been editing the pro football section for the Sporting News. There were also a couple of hockey guys. I did baseball. That meant that each of us wrote only about what we knew best, which helped a lot. More papers should try it. It also helped that the editor then, Cliff Froelich (now doing film festivals) seldom edited anything beyond copyediting, and never for content. So we were all free to say whatever we thought. When Tony came to St. Louis, I realized that my preseason preview was a disaster because it bore no resemblance to a Tony team. Having had experience figuring out Whitey Herzog's approach to a roster, I took a long look at what Tony DID, at least as much as what he said. That's the luck that led me to the pool system. So, blame it all on Whitey and Tony. They've been blamed before.

So, what is the chance that at least three of the six have seasons in the top quarter of what they could have?

The answer, easy to calculate, is 94%.

Sorry Brock, but that answer is dead wrong.

The chances that NONE of them will have a season in the top quarter is 18%. That's .75 to the 6th power. There is roughly a 17% chance that at least three of them will.

Or you're making some very funny assumptions about those distributions because I can't figure a way of getting .94%. The best I've done is that you get 96% chance of at least three of them being better than their BOTTOM quartile of projected performance (which is the same as the chances of at least 4 of them being in the top quartile).

And that's before we get into assumptions about how/when the manager decides who gets the playing time.

As to your "pools" ... I'd say the "bat pool" is really 1B/LF/RF (and DH in the AL) as a good-hitting CF is something of a luxury. CF belongs in a glove/bat pool with 2B and 3B although with the possible exception of the backup, there's no positional flexibility in that pool. C and SS form the glove pool and you need a "legit" backup for each. And as of course you know, truly good teams are often characerized not by a good bat pool but by good-hitting glove/bat and glove pools. The Yanks dynasty with Posada, Jeter, Williams and Knoblauch/Soriano was a classic example.

Zombified -

Thanks to you, too. You're doing exactly what I hoped readers would do - try to apply this to their own teams. Note that the Marlins, in your breakout, only have two batpoolers who can play center. That limits their flexibility and synergy far under what Tony has in St. Louis. Also, the glove pool seems to have only one backup who can play either short or second. I have doubts about that. Are you sure that neither of the third base backups can play anywhere else? If not, this team is in trouble if Uggla and Ramirez get hurt at the same time. Of course, if I read the team right on slight analysis, they're in trouble if those two get hurt no matter who the backups are. BTW, this kind of team can surprise, and you can see it coming in the preseason; it's happened before, and in Florida. If you have a good Glove Pool and some pithing arms, but a Bat Pool that can't hit a lot, it's always easy to improve the team a lot if you're willing to spend even good money rather than great. You need outfielders who can hit; there's always a free agent or seven out there. Like Gary Sheffield, who jelled a team just like that into a winner. If the pitchers and Hermida come through, they're just about one bat away, if the pitching is there.

The Bat Pool: First Base and the Outfield

I thought the Bat Pool was where Bruce trains for team-ups with Aquaman.

On the pools as Brock's constructed them, the key (as Brock alludes to) is whether your 4th OF (i.e. your best non-starting bat and quite possibly the 4th or 5th best hitter on the team) or one of your starting LF/RF can handle CF. This is what gives you the ability to mix and match based on L/R, health, etc. without losing much offensively or defensively. But really that's just another name for bench depth. Your 4th OF almost always gets at least 300-400 PA, the backup C gets about 150-200, the main backup MI gets about 300, etc. Any projections which don't build in those sorts of playing time assumptions are going to lead you astray. The key with projecting the Cardinals last year (and I suppose this year) is that it didn't really matter what PT assumptions you made about those guys, you'd get a pretty "average" overall projection for your "starters" and above-average for your "backups."

Brock's right that the modern bench is different than the past. (I think he probably oversells the extent to which other folks are projecting based on 8 starters and 8 backups.) The typical bench now is one backup C (who usually can't hit), one backup SS (who usually can't hit), one backup OF who can play CF (and usually can't hit). That leaves two spots for guys who can hit except that often one of those is a swing IF/OF (who usually can't hit). LaRussa almost always has one of those swing IF/OF players on his roster but they almost never can hit (Schumaker this year being an exception). That leaves you with one backup corner player who can hit.

Good teams will often have the OF flexibility mentioned above or someone like Mark DeRosa who's an average bat at 2B/3B and won't embarrass you in RF (and who had an excellent year last year).

In the AL, things are even more constrained. If you carry a real DH, you've got only 4 bench players with backup C, backup SS and backup CF eating up 3 of the slots (and not hitting) and you'd still like a bench IF/OF type. On the plus side, you're rarely pinch-hitting. Still, this is why lots of AL teams don't carry real DHs but instead rotate their bench players through that slot (or put their bench players in the field and give the starter a "day off" in the DH slot).

Are we getting a full series of team previews? I hope so, I always enjoy them. I had given up on that happening this season...

Walt Davis says, "The answer, easy to calculate, is 94%.

Sorry Brock, but that answer is dead wrong."

Walt, you're right. Apparently the calculation is more complicated than I thought. I'm going to do it again and see what I did wrong. But .75 to the 6th is, indeed, about 17%, so it's obvious I have this wrong. And I have a degree in math. How embarrassing.

As to what spots belong in what pools, I'm looking at it from the point of view of a manager. Managers don't lump center fielders in with third and second basemen, They lump them in with the other outfielders because of the differences between outfield and infield defense. In terms of defensive spectrum, you are absolutely right, but that's not how managers look at it. There are, as far as I know, no third basemen / center fielders. There are a lot of center fielder / left fielders. There are first basemen / outfielders only because first base is considered to be so easy to play. In other words, what drives my pool divisions is the defensive synergy among various spots. First and the outfield have defensive synergy. Second, short and third have it, too. Actually, first base has it with about anything, but almost every manager looks for bats at first, so it goes into the bat pool.

In Yankee terms, Jeter doesn't belong in a pool with Bernie Williams because Jeter was never gong to play center, and Williams was never going to play short. No synergy. Posada was only going to play catcher or first. No synergy with either Jeter or Williams. So I start with defensive synergy and then look for what qualities the manager expects out of which synergy group. The synergy group of outfielders and first base are expected to provide bats more often than gloves. The opposite is true of the infielders.

DCW3 says, "I thought the Bat Pool was where Bruce trains for team-ups with Aquaman."

As a comic collector since 1956, I say: Haw! Other than that, however, I absolutely refuse to discuss Batman and Aquaman's water sports.

this pool analysis? i don't like it -- I LOVE IT! a good readable article, and a neat premise. this stuff is why i like bbtf.

This was a good article. I think it's a little biased but very good.

I saw Ludwick play several times at Toledo and if anyone had told me the guy would slug .600 in the big leagues I would have said that's absurd. He looked all the world like a 25 homer guy in a good season but with the potential to have the strikeouts push him to 4th outfielder status. Sure he had power. He also struck 170 times that season in 500 at bats and in like five games I think I saw him strike out a dozen times.

He has reduced the strikeouts a tad and ramped up the walk a bit which is what makes it so interesting. Which may seem minor but actually has HUGE significance. Because he works the count in his favor just often enough where the power can have an impact.

Toledo is a tough park which is why even at .260 in the minors I figured he could keep a job if the K's didn't eat up his at bats.

Ludwick's season can make sense if you look at his time at age 21 and consider the development curve. It's a tribute to him that the chronic injuries didn't eat away his original skill set.

Walt says, "The best I've done is that you get 96% chance of at least three of them being better than their BOTTOM quartile of projected performance"

I found my old calculation worksheet, and that's exactly what I calculated: the better-than-bottom-quartile synergy. I apologize to everyone. Walt, since you probably have a stats package that can do this without thought (or you can do it in your head, which is beyond my abilities at combinatorial math), what IS the chance that, of a population of six players, at least 3 of them will turn in a year that is in their personal top quartile? No hidden assumptions; just the top quartile. That's what I was trying to do last year. I really want to know, and I'm afraid I'll screw the calculation up again. I know you won't.

And thanks SO much for catching this early, before I embarrassed myself all over town.

I saw Ludwick play several times at Toledo and if anyone had told me the guy would slug .600 in the big leagues I would have said that's absurd. He looked all the world like a 25 homer guy in a good season but with the potential to have the strikeouts push him to 4th outfielder status. Sure he had power. He also struck 170 times that season in 500 at bats and in like five games I think I saw him strike out a dozen times.

He has reduced the strikeouts a tad and ramped up the walk a bit which is what makes it so interesting. Which may seem minor but actually has HUGE significance. Because he works the count in his favor just often enough where the power can have an impact.

Toledo is a tough park which is why even at .260 in the minors I figured he could keep a job if the K's didn't eat up his at bats.

Ludwick's season can make sense if you look at his time at age 21 and consider the development curve. It's a tribute to him that the chronic injuries didn't eat away his original skill set.

Most of the pre-season stuff I saw argued that his performmace is not sustainable, but that he will still be good 280/350/500.

Robinred says, "This was a good article. I think it's a little biased but very good."

First, thank you! Second, I'm always looking to improve my work. Aside from having Walt Davis check all my math from now on, I always want to know when I've been guilty of bias. So, could you tell me where you see the bias? I'm not questioning you; I'm questioning me. I can be guilty of bias, I just want to go down fighting it. Thanks in advance, - Brock

The benefit of having a great manager is that the club can pick up guys of minimal market value knowing that there is a high probability the manager will get more out of them than anyone thought possible.

It's one of several things about Tony that drive fans of rival teams pretty crazy. Take Khalil Green. He's had some moments, but he's not a great player by any stretch. He had his "big" year at age 27 and now at age 29 one would expect him to be at best a stopgap. Little pop but with no real control of the strike zone undermines his offense. Defensively you can see why he plays.

Everyone in the NL Central knows that dollars to donuts Greene will play like Jeff Blauser in a good year and show up with plus defense, 25 homers and fifty walks out of nowhere. Why? How the f*ck should I know? That's just what happens on a TLR team.

Harveys - Have you ever read my "iambic development" theory anywhere? Essentially it says that most (not all) young players alternate better and weaker seasons until they hit ages 26-28. Greene fits this mold. He had a hot age 27, then a down 28. He ought to be up for a good 29. On the general point, you're right. Having listened to hours of Tony and Dave Duncan talking about this, their main skill seems to be identifying the types of players they can help. Dave works great with veteran pitchers who have only rising fastballs. He teaches them the sinking one and preaches "pitch to contact." Didn't work with Anthony Reyes, who wouldn't give up his riser, but it worked great with Kyle Lohse. This year, already, Jason Motte blew a save early by shaking off Yadier whenever he called for a sinking fastball or a slider. He hasn't closed since. Dave wants him to learn the lesson: You can't just throw the heat past major league hitters. Throw the slower thing that sinks, so they hit it on the ground. Tony, too, is able to identify players who have one big weakness, and its one that Tony knows how to fix. Most managers do this, but Tony seems to have a large range of problems he knows how to get rid of. With Greene, it's probably a help that Tony was a hot glove shortstop and knows the position, although it's the bat that needs help. That is, it's not so much a talent for fixing anyone; it's a talent for identifying the people they can fix. And if I were not a Cardinals fan, it would drive me crazy, too. - Brock

I apologize to everyone. Walt, since you probably have a stats package that can do this without thought (or you can do it in your head, which is beyond my abilities at combinatorial math), what IS the chance that, of a population of six players, at least 3 of them will turn in a year that is in their personal top quartile? No hidden assumptions; just the top quartile. That's what I was trying to do last year. I really want to know, and I'm afraid I'll screw the calculation up again. I know you won't.

I'm not Walt, but I am a combinatorialist, and the answer (showing cryptic work) can be obtained via:

1 6 15 20 15 6 1
1 3 9 27 81 243 729
-------------------
1 18 135 540 ... ... ...

1+18+135+540 = 694, /4096 = about 17 percent, as Walt noted.

Brock:

I wrote the question of why and the response just to be funny. I have gathered over time Tony's approach. What I was hoping was that he would be shut out by the new leadership in offering input, get stuck with guys he didn't want, give them the cold shoulder in rebellion against management, try to win with "his guys", and if things went bad leave town. Preferably for another team. Out of the NL Central.

Sigh............

Not to take a cheap shot at the Cubs, but to offer help, the failure to do this has hurt the Baby Bears more than once. They use an old-school minor league approach, where the AAA team is treated as its own roster with its own pennant to win. The Cards realize that you simply need more than 25 major leaguers to fill out a season. So they use AAA as a way to stash a few extra backups. Over the course of the season, that helps. The Cubs always seem to have more front line than the Cards, but the Cards always seem to overperform while the Cubs wimp out. The failure to max out AAA usage is one big reason.

A couple thoughts with recognition of my Cub fan bias.

First, given the focus on 2008 throughout the rest of the piece, I think it's incorrect to suggest a "wimp out" from the Cubs. If that's a pile-on for the postseason follies, so be it. But it's rather silly to suggest the Cubs somehow underperformed their season projection last year, or any recent year save 2004.

Second, I've seen the Cubs having a tendency to stash players with MLB experience like Michael Wuertz, Matt Murton, Ronny Cedeno and Micah Hoffpauir in AAA for an emergency. Probably others if I gave it some thought. But perhaps they do this less than St. Louis. I'm not sure, but anecdotally, it doesn't seem right to praise the Cards for maximizing extra talent in AAA and damning the Cubs for mismanaging this potential extra resource. In fact, this year sees Jeff Samardzija, flush with options, available as a capable insurance policy for the popularly predicted Harden injury, whether he takes a spot in the rotation or replaces Heilman in the pen. Though I can't think of a lot of other AAA guys that may be called upon--Kevin Hart or Sam Fuld maybe.

This is an interesting method. I'll do the Orioles. Not sure what to do with Wigginton and Freel - Wigginton is a platoon DH/1B, but he is also a backup 3B... Freel is a platoon LF, but he is also a backup 2B/3B...

Bat Pool:
Huff 1B
Scott LF
Jones CF
Markakis RF
Pie LF/CF
Freel LF/CF/RF
Wigginton 1B/LF

Glove Pool:
Roberts 2B
Izturis SS
Mora 3B
Andino SS/2B
Freel 2B/3B
Wigginton 3B

Catcher Pool:
Zaun C
Moeller C

Starter Pool:
Guthrie
Uehara
Hendrickson
Simon
Eaton
Bass

Closer Pool:
Sherrill
Ray

Middle Reliever Pool:
Jim Johnson
Sarfate
Walker
Baez

oh, here are the notable players stashed in AAA/DL:

Montanez LF
Riemold LF/CF/RF
Salazar 1B
Wieters C (duh)
Albers middle reliever
Bergesen starter
Hill starter

You were paying attention to Ryan Ludwick's swing and how hard he hit the ball? In 2007??!!! WOW, do YOU follow baseball!

Ludwick had a couple of homers in 2007, the one on 6/17 in Oakland being the most memorable for me, where he demonstrated remarkable bat speed. I didn't see him slugging .591 or anything, but I would've bet the over on his ZIPS.

The Cardinals pretty clearly need another RH OF. Mather's suffering from a hand injury that could cripple him for 2009, TLR hates Barton, Stavinoha's not that talented, and Shane Robinson is 5'8". Start the Lastings Milledge rumors here, but until then it would help if TLR would let Rasmus be an Endy Chavez-type against LHP rather than playing the polar bear in left field.

Stevens -

I was talking about the last several years of Cubbie AAA practice. Yes they do have some vets down there, but nothing like the Cards store away, and it doesn't seem to be so systemic. The Cards stash someone for every injury contingency, not just for the predictable ones. That's probably the main difference. . You were able to come up with three for this year, while I came up with 5 Cards without looking at backup starting pitching. I would actually like to see the Cubs win one. I mean, it's been a century now, and a couple of my good friends are Cub fans who get depressed every October. When I look at the Cubs, I see the lack of a really extended roster as a weakness that hurts them. Wanting them to win if the Cards don't, I posted up the one thing I think I have a handle on that would help. It's the best I can do. I do protest that I am not trying to slam the Cubs. I'm trying to help, within my limited abilities.

Greenbakc says, "The Cardinals pretty clearly need another RH OF."

First, I'm not at all sure that I agree. I think the Cards have enough righty bats that they don't need any help there from the outfield. Having said that, I hasten to add that there must have been some good reason why Brandon Ryan got spring work in the outfield. For him to be the backup righty OF sounds like Tony to me.

Jason Kendall #6 did the combinatorials on top quartiles. Thanks! I actually understand the method you're using (1 chance of all 6, 18 chances of 5, 135 chances of 4, 540 of three, out of 4096 altogether). My problem with math is that I'm MUCH MUCH more an applied math guy than a theoretical one. You give me a formula, and I can "plug and crank" real good. When I did the calculations last year, I was trying to re-invent the formula for that combinatorial, and I messed it up (actually, I used .25 everywhere I should have been using .75 and vice versa). Just the numbers you listed makes the theory clear. I was going to ask Walt, and he'll likely read this, but what I really need is a good reference book on combinatorial methods (or probability and stats in general). I don't need a textbook on theory, though. I need an applied book that has all the formulas there along with a good Table of Contents and index, so I can find the right formula quickly. I can do this sort of thing: Just 4 years ago, I got a discalculic and dyslexic through a stats class by being able to read his textbook and then apply the formulas. But when I have to start from ground zero and develop the formulas from basic theory, I'm not so good. So, do you know of a good book for me? I have no ego about this. If there's a "for Dummies" that works, that's fine. If it takes something advanced from CRC, that's fine, too. And if you don't know of one, well, someone will. I now know that I need to get a book to work with. Thanks in advance - Brock

Harveys says (say?), "I wrote the question of why and the response just to be funny."

It was funny. I had a good chuckle. But it also brought up a good point about Tony. I'm sure he'd try to help you out in Milwaukee, if you were still in the AL, but he doesn't have Ted Simmons to trade and you're in the same division now. We do the best we can. I even spend money at the Safe House so there's more for the Brewers to spend on free agents.

re paying attention to Ryan Ludwick's swing and how hard he hit the ball? In 2007??!!! WOW, do YOU follow baseball!

Seriously? Luddy had a .880 OPS the last 4 months of the season, with 14 HR's in 275 PA's. He basically had a tremendous spring (1.000+ OPS), a tremendous first month in AAA (1.022 OPS), 3 bad weeks in MLB in May, and then an outstanding finish.

I think the Cards have enough righty bats that they don't need any help there from the outfield.

I don't know what "enough" is, but when you've got what are probably low 700 hitters starting in the OF every night an LHP starts, then you're at a disadvantage. It doesn't help that their third- and fourth-best RHBs are probably Khalil Greene and Yadier Molina.

Very good analysis. The only nitpicks I have with the LH/RH batting analysis and 3rd string catcher.

This team will likely suck against LH pitching, even with Pujols and Ludwick. The problem is you have Schumaker, Duncan, Ankiel, and Rasmus, all lefties, none of them hit lefties well, and at least 2 or 3 of them will be in the lineup vs. LHers. Schumaker in particular really needs a platoon partner, but Thurston is also left handed, and neither Ryan or Barton can hit a lick.

And Pagnozzi could conceivably come up for a couple weeks if LaRue were injured and watch Molina play every single game, but he's not even a good AA hitter, and if Molina goes down, you can bet that Brian Anderson would be the one called up, not Pagnozzi.

There's yet another Brian Anderson? Ugh.

This on the team that already has Brendan Ryan, Brian Barton, and Brian Barden.

"This on the team that already has Brendan Ryan, Brian Barton, and Brian Barden."

I was watching a Cards game with a friend of mine, when I heard that Brian Barton was on deck. I said that Barton was a dread-locked black outfielder who wasn't drafted due in part to his an aerospace degree. Some white guy named Brian Barden ended up batting...

I looked like an idiot.

Brock,

I kinda suck at combinatorics too but I just used the binomial distribution for this ... and that uber-fancy stats package Excel. :-) If memory serves, the PDF of the binomial is given by:

(n choose x)*p^x*(1-p)^(n-x)

To get the CDF, just sum that over x=0 to whatever. For your problem, sum from 0 to 2 with n=6 and p=.25 and that gives you the chance of 2 or fewer hitting their top quartile. One minus that of course is the chance of three or more. For quick and dirty (but inaccurate for low values of n), just figure the expected number (mean) as n*p and the SD as sqrt(n*p*q). In this case, that gives you 1.5 +/- 1.1, putting 3 more than one SD above the mean. For low values of n, the normal isn't a particularly good approximation for the binomial but it will still get you in the ballpark and, for a normal distro, one SD or more above the mean is about 17%.

While your numbers were way off, your point (in its way) is still valid. The Cards were not almost guaranteed to have a way-above average OF, but they were nearly guaranteed not to have a big hole out there. In other words, start with 6 average OF and the chances that at least 3 of them will be better than their 25th percentile (i.e. they won't suck) is nearly one. In fact, you'd expect 3 of them to be average or better.

Which brings us back to the pools. I like the idea in general and I understand your reason for not lumping CF with 2B and 3B. But regardless of how managers think, most teams are lucky to have one average CF much less two. For the Cards last year, they were fortunate enough to have two guys who were above-average as CFs. But a lot of teams are carrying a Joey Gathright or Jerry Owens as their backup CF ... not because they aren't smart enough to build flexibility but because there aren't that many average CFs out there.

There's also the school of thought that OFs are pretty much interchangeable -- move an average hitting, average defensive RF to CF and you get an above-average hitting, below-average defensive CF with those two essentially cancelling themselves out. Under that theory, essentially every team has this flexibility and it's purely a matter of the quality of the various players.

I guess at some level I just think you're over-complicating things. Give or take, every position gets 700 PA. While it's not guarantee, you figure each starter (assuming you have a full-time starter at that position) gets about 500-600 PA. I'm not sure how many folks are really aware of this, but in any season for an NL team, they only average 4.5 hitters who qualify for the batting title (500+ PA). Of course you pretty much rule out C to begin with.

Anyway, the point is you can pretty much count on your 4th OF getting 300-400 PA and your backup CF and two backup MI getting about 200 and your backup C getting about 300 not to mention the various minor-leaguers, etc. That's about 1200 PA, or nearly 2 full-time positions, coming from your bench. The Cards' strength was their depth allowed them to give half those bench PAs to average-ish OFs (not to mention Miles hitting pretty well) but it's not so much where that "flexibility" came from -- if they'd had two average-hitting MI backups or one MI and one OF, they'd be just about as well off.

So yes, it's silly to evaluate teams based just on their starting lineups. But an evaluation of a team that's basically "their starters project as X wins better than average but their bench is Y wins worse and they've got no real prospects who might step up" is fine. Now ... for a team preview, the extra detail is useful but I'm not sure that's much more than saying something like "the Cards are in much better position to handle an injury in the OF than one in the IF."

I suspect the more important issue about flexibility is L/R platooning capability ... or potentially defensive platooning (based on whether your starter is GB or FB heavy) ... or other forms of platooning. A good platoon can quickly turn two below-average OFs into 800-900 PAs of average production.

Anyway, I guess it's more semantics than anything.

Walt -

Thanks for the math. I would never have thought to trot out the binomial theorem. BTW, I'm guessing that your primary background is theoretical. I could follow your discussion, but just barely. It was almost too abstract for me. In fact, the estimated SD stuff is too abstract; I know where you went with that, but not in any detail how you got there. I'm a plug and crank guy, really, and would have had a degree in computer science if such things had existed at Vanderbilt in the 1960s. Also, as in my post above, can you recommend a book? I've had enough of making mistakes because I'm too lazy to ask for help and get a good resource that fits my personal approach and talents and then learn how to use it. I'm gonna get me a book or two and stop embarrassing myself instead of having to apologize.

I do agree that a lot of what passes for a disagreement here is semantics. What I'm trying to do with pools, aside from mirroring manager-thought, is to combine looking at depth and looking at flexibility. You're separating them and demonstrating that one can simulate the other. That's true, but it's not the same as having both. With both, you end up dodging the weak backup problem that you pose. Your comment,

"For the Cards last year, they were fortunate enough to have two guys who were above-average as CFs. But a lot of teams are carrying a Joey Gathright or Jerry Owens as their backup CF ... not because they aren't smart enough to build flexibility but because there aren't that many average CFs out there."

is dead on. The fact that the Cards actually had five guys like that was what led to the massive underpredicting of the 2008 edition. No one noticed because of the focus on separating starters from backups. What I was pointing out was that the Cards, if you looked at who had what ZIPS projection, had five CF starters, not a starter and some backups. The individual starters weren't very good compared to other starters, but there were so many of them that at least some had to come through.

The next paragraph in your post is really interesting to me. I've wondered for years whether a team would be better, assuming the personnel, to put a big bat outfielder at second or even shortstop because the bat gain would outweigh the glove drop, even though the glove drop would be severe. No one has ever tried it, but you could argue that, say, the Red Sox could have moved Youkilis to short rather than first. They would have lost a lot of glove compared to a real shortstop, but they would have gained so much bat.... I don't know. I would dearly love to find out, but getting a real baseball manager to try the experiment is likely very unlikely.

You are also very correct about the number of full-time starters. I've also noticed that it amounts to about 4.5 guys per team who qualify for the batting title. The interesting thing to me is that this number hasn't changed much over time, until you get back to the 19th century. It's always been about 4.5 qualifiers per team. What I've wondered was if that info could be used as a cornerstone in trying to estimate a replacement rate. The way you've used the info in your post - counting up the expected PA for the various backups - comes close to looking at replacement rate, or at least "backup rate."

In short, I think you're right. We're pretty much on the same page. We're pretty much using the same info. You're looking at it more generally, whereas I had a question about a specific and weird team to answer - what were the Cards going to get out of their outfield and why? You thought about things like complex platooning, where you are certainly correct.

- Brock

Brock,

Are you sure that neither of the third base backups can play anywhere else? If not, this team is in trouble if Uggla and Ramirez get hurt at the same time.

Helms and Cantu back up 3rd, Bonifacio can back up 2B (his natural position). Amezaga is the only SS backup.

But yes, this team is royally screwed if Uggla or Ramirez go down. Not only is that their starting SS and 2B, it's also the Fish's 3rd and 5th-spot hitter. Not easy to replace.

Looks like Chris Carpenter is headed to the DL once again, though at least he hurt himself batting and not throwing.

I feel bad for this guy, he's so talented and just can't seem to catch a break lately.

I would never have thought to trot out the binomial theorem.

Somebody needs his math degree revoked. ;)

Excel has a nice function to deal with this: BINOMDIST.

BINOMDIST(3,6,0.75,TRUE) gives the cumulative (the TRUE parameter) probability of 3 successes (or fewer) in 6 trials where the probability of success is .75. This is analogous to 4 successes or more where the probability of success is .25, the situation you are interested in.

Just following up on my last note (because I can't seem to find an edit button), a more deliberate way of using the Excel function to answer the question "what is the chance that at least three of the six have seasons in the top quarter of what they could have?" is by instead examining the question "what is the chance that 2 or fewer of the six have seasons in the top quarter of what they could have?" and then subtracting that from 1.

100% - BINOMDIST(2,6,0.25,TRUE) = 100% - 83% = 17%.

There is an 83% chance that 0, 1 or 2 of the 6 will not exceed their 75th percentile.

Chuck says, "Somebody needs his math degree revoked. ;)

Excel has a nice function to deal with this: BINOMDIST."

Yeah. I know. The odd thing is that, if someone had told me that this was done using the binomial theorem and that Excel has a function for it, I'd have gotten it right. I understand the concept of figuring the probability of the opposite of what you're looking at and subtracting from 1 (100%). If you give me a formula and tell me to plug and crank for the question at hand, I'm good. The problem here was that I tried to figure out what formula to use from ground zero, meaning that I was trying to duplicate the theory. That's not my game. I need books with formulas in them. The odd thing is that I can do what I did just a couple of years ago with my discalculic friend. Every week, this poor guy would meet me at a coffeehouse, and I'd do his homework (permission granted by the professor). My method was to go through the chapter in his textbook, figure out how the formulas there were used, and do the problems. That would take me maybe a couple of hours for the whole chapter's homework problems. But if you'd given me the same problems with no chapter to look up the formulas in, I'd have been helpless. The theory is beyond me. That's why I talk about my math degree being really an applied math degree. I'm not a theorem prover. I'm a word problem solver, and for that, you need the formulas in hand. It showed up in college. In the applied math courses (Engineering School), you got to bring books to tests. They didn't care if you had memorized the formulas. In A&S;School math, the tests are no-book. That, in a nutshell, describes the difference between how I think and how pure mathematicians think. I need open book, and I've been dumb enough to end up with no stats and probability books in my house. I need a book recommendation. Then I can stop explaining why I mess things up when I have this degree that should apply. In the meantime, such good-natured humor as you posted is more than welcome. I'm laughing at myself, and see no reason not to share. Have a good time.

I saw Ludwick play several times at Toledo and if anyone had told me the guy would slug .600 in the big leagues I would have said that's absurd. He looked all the world like a 25 homer guy in a good season but with the potential to have the strikeouts push him to 4th outfielder status. Sure he had power. He also struck 170 times that season in 500 at bats and in like five games I think I saw him strike out a dozen times.

He has reduced the strikeouts a tad and ramped up the walk a bit which is what makes it so interesting. Which may seem minor but actually has HUGE significance. Because he works the count in his favor just often enough where the power can have an impact.

and to be honest Harvey that is what I saw, the first half of 2007 Ludwick didn't look to hot to me, but when he started to earn more playing time, his patience increased, he seemed to be working deeper into the counts, and the noticeable hole in his swing was reduced. That is when I became a fan because it looked like he fixed something, and it looked like it would stick.

And Pagnozzi could conceivably come up for a couple weeks if LaRue were injured and watch Molina play every single game, but he's not even a good AA hitter, and if Molina goes down, you can bet that Brian Anderson would be the one called up, not Pagnozzi.

I don't have that impression, if Molina goes down, Larue gets slotted as the starter and they'll bring up Pagnozzi, at no point has Anderson impressed Duncan/TLR that he can handle a pitcher. Heck the cardinals have pretty much flat out stated that Pags will be the first called up.

Brock:

No, I'm a numbers cranker. I didn't derive the formulas for the mean and SD for the binomial, I just have them memorized because I use them a lot (and they're not hard to remember). Sometimes I work back a bit to the theory and most of the algebra doesn't throw me. So it's a lesson to me on how well I explain things that you thought it was too abstract when I thought I was giving cookie-cutter. :-)

So here's another attempt at cookie cutter...

Anytime that you can view a problem as "how often will event E happen when I roll this die N times" or "in N rolls, how likely is it I'll get at least X Es", the binomial is likely to give you at least a good approximation (and under the right circumstances the correct answer). The key assumptions are that each roll of the dice has the same probability of resulting in E and each roll of the dice is independent (i.e. what you roll on your first toss doesn't affect the probability of your second roll resulting in E).

Now, the mean of the binomial is N*p (see below if you really want to know one way to get there). The variance is N*p*(1-p). You can just look those up. The standard deviation is just the square root of the variance.

So in your case, you have 6 OFs. You can think of those as 6 rolls of the same die or 1 roll of 6 identical dice, doesn't matter. You're interested in how often they'll finish in a given quartile so you think of it as a 4-sided die. And the question becomes "in 6 rolls of a 4-sided die, how likely am I to roll at least 3 fours?" You could use the formula for the PDF and the CDF I gave earlier to get the precise answer.

However, one of the nice things about the binomial distribution is that, when N is large enough, it converges on the normal distribution. A nice thing about the normal is that it's pretty easy to get the probability of events using the mean and the SD using a z-score. If you don't know how to calculate those, just look them up. Now 6 is not large enough a value of N for the normal to be a really good approximation to the binomial but then you may not need a real precise answer. The mean here is 1.5, the SD is about 1.1, so 3 would be more than one standard deviation above the mean. Now the normal is also my friend and I know in the normal that the mean +/- 1 SD covers about 2/3 of the distribution so exceeding the mean plus 1 SD happens about 16-17% of the time. The mean +/- 2 SD is (approximately) our good friend the 95% interval. (Note this shows why the normal isn't a good approximation of the binomial for small N -- for N=6 and p=.25, the mean minus 2 SD gives you a negative number while the actual minimum count is zero of course.)

The binomial can be handy for lots of q&d;calculations in baseball. Suppose a team is a "true 500" team. What is the 95% confidence interval on their wins? Well, the expected value is of course 81 (.5*162). The variance around that is 162*.5*(1-.5) or roughly 40. The square root of 40 is about 6.5. So they'll usually fall within 74.5 to 87.5 wins with about a 17% chance of exceeding 87.5. The 95% confidence interval is 81 +/- 13 wins!

Suppose a batter is a "true 350 OBP" hitter. Over 600 PAs, what is his likely range of outcomes? .35*600 is 210. 600*.35*.65 is 136.5 and the square root of that is 11.7 so let's call it 12. 12 is .020 of 600 so such a better has about a 2/3 chance of posting an OBP between 330 and 370.

Now, those are both approximations. Remember I said two assumptions of the binomial is that the probability of the event is constant for each roll of the die (or flip of the coin) and that each roll is independent. The first condition is obviously violated -- a team's chances of beating the Nationals is greater than their chances of beating the Cubs; a batter's chances of reaching base against Ponson are better than against Lincecum. The second condition may be violated if hot/cold streaks or momentum really exist but I think it's fairly safe to assume independence.

One way of getting there ... each toss is what's called a Bernoulli trial, an experiment where there are only two outcomes: 0 (non-event) or 1 (event). The event occurs with probability p so the expected outcome of a Bernoulli trial is p -- if you flip a coin, the chance of a head is 50%; if you roll a 6-sided die, the chance you roll a 3 is 1/6. The variance of a Bernoulli trial is given by p*(1-p).

The binomial is a series of independent Bernoulli trials with constant p -- 100 tosses of a coin. Each toss produces a 0 or 1. Because of the independence and constant probability, each toss has the same expected outcome. To get a little fancy, let's make each toss it's own random variable called T1 to T100 which in any give set of 100 tosses take on the values 0 or 1. Note the number of events in 100 flips is the sum of T1 to T100. Using some other stats theorems, under independence:

E(T1 + T2 + T3 + ...) = E(T1) + E(T2) + E(T3) + ...

The expected value of each toss is p, so the expected number of successes in 100 tosses is 100*p.

Moving on to the variance, under independence it's also true that:

V(T1 + T2 + T3 + ...) = V(T1) + V(T2) + V(T3) + ...

The variance of each toss is p*(1-p) so the variance of the number of successes in 100 tosses is 100*p*(1-p).

And back to the pools ...

What I guess I object to is the claim that managers look to CF as one of the "bat" positions. Managers generally don't look to CF for a bat, at least no more than they look to 2B and 3B for a bat. And while there's clearly more defensive "exchangability" among the OFs than among 2B/3B and CF, few managers are willing to put a poor defensive CF out there no matter how well he hits (unless it's a star like Griffey). You could say the same for the 2B/SS/3B pool in that managers look for a lot more offense out of 2B and 3B than SS and there are lots of 2B and especially 3B that managers would never let play SS. You've also got the problem that there are several players these days cutting across the IF and OF pools (DeRosa, Miles, Bloomquist, etc.). But there's no simple clean solution.

One way to go is to be more detailed. For example, for the purposes of projecting the Cubs, I would break it down something like this, assuming 700 PA per position:

LF: Soriano 500, Johnson 100, Hofpauir 50, Gathright 50
CF: Fukudome 450, Johnson 200, Gathright 50
RF: Bradley 400, Hofpauir 100, Johnson 100, Miles 50, Gathright 50

Now, on the IF, the Cubs look pretty screwed to me because the only backup is Miles. I suspect that will change as the season progresses -- and has Johnson started taking groundballs yet? Anyway, while this isn't too realistic, this is what the Cubs have to hope for at the moment on the infield:

3B: Ramirez 600, Miles 100
SS: Theriot 600, Miles 100
2B: Fontenot 550, Miles 150
1B: Lee 600, Hofpauir 100
C: Soto 500, Hill 200

So that makes the Cub bench:

Johnson 400, Miles 400, Hofpauir 250, Hill 200, Gathright 100

Now obviously some AAAs will get PAs, a healthy Lee will start nearly every day and get nearly 700 PA, some starter will get injured and have just 250 PA, I haven't included PH appearances yet. All told, the Cub bench probably won't get quite that many PA. By the way, one key to the Cubs' success last year was that they had 7 players get 500+ PA (including Soto and Soriano despite him missing a good chunk of time). The only position they didn't was CF where they were running a platoon.

This is probably one of the remaining challenges for team projections. For an individual player projection of a starter, it probably makes sense to project them to 500+ PA. But in terms of how that's going to work for a team, as we know, only about 4.5 of them are going to make that threshhold. Team projections probably shouldn't use the straight individual projections of the individuals on the team but rather build in a range of playing time assumptions. In theory you could do this by having baseline playing time assumptions but then simulate injuries, etc. in a realistic fashion.

This is a bad defensive team with mediocre pitching and a mediocre lineup outside of Pujols and Ludwick. Without Pujols I don't even think this is a .500 team on paper. I'll take the under, something like 84-85 wins and only that high because of the strong start.

This is a bad defensive team with mediocre pitching and a mediocre lineup outside of Pujols and Ludwick. Without Pujols I don't even think this is a .500 team on paper. I'll take the under, something like 84-85 wins and only that high because of the strong start.

Say what you want about the pitching, but the Cards have a good lineup - they had a top offense last year, and they're leading the league so far this year. The defense has alwas been above average under TLR, and is off to a good start this year. They won 86 last year, and it's unlikely they'll be worse - 90 wins is looking pretty good right now.