Good stuff. I like the graphs.

It matters which Pythagorean formula you use. I took a look at this:

In fact, the offense that contributed the most wins over the average distribution was that of the 2003 Dodgers' team, about 5.5 wins more than you would expect given their average number of runs scored.

Using the standard exponent of 2, that's the result. But using an exponent of 1.81 (which is suggested by Pythagopat if you have the Dodgers playing a league-average, 4.82 rpg team) the 5.5 win advantage drops to 3.3 wins. That is significant.

Of course if you are going to come up with some sort of offensive winning percentage, everything needs to be park-adjusted. But for calculations like this I doubt that a park adjustment is necessary.

It looks like it should be possible to come up with a formula for the distribution of RPG. That could be used to come up with a more accurate Pythagorean formula, though it probably would be only a minuscule improvement over the ones we have now.

I would be interested to see if Pythagorean results matched reality better if you threw out all blowouts (say, games decided by 9 runs or more for or against.)

The distribution presented by Dave pretty much follows my run distribution model. You can download that program from my site (last link on home page). Set the control value to .7667, which is the norm for 1974-1990. The norm for 1901-2000 is .761.

What's interesting with my distribution is that you can change that control value to .70, or .85, and that will model different styles of teams' run distribution pattern.

(This control value talk means nothing unless you download my program.)

Tango, thanks for the program. As usual, it's super, and it's great of you to make it available.

I read something interesting in the attached html pages that I hadn't thought of before (quoting the Jim Baker email) that a single run scored is worth more than a single run scored in a batch (three-run home runs, etc) due to the relative win contribution of runs 2-5. Does this make sense to people? I assume that it roughly translates into the PythagoPat (or whatever!) exponent, and it would probably play out that way in Win Expectancy too.

I don't think there's much to it. I've shown the change in RE (run expectancy) and change in WE (win expectancy) by event, and that ratio (runs-to-wins) is pretty stable among the big components (walks, hits, hr). Perhaps there is something to it, but, I doubt that there's much.

Yes, that's where the discussion breaks down: leaping from the relative impact of single runs to single run strategies.

If you play for one run, as Earl Weaver would say, you'll only score one run. So single runs may be worth a bit more than individual runs within a batch, but the strategy used to score single runs may give up too many runs from the batch to be worth it.

So single runs may be worth a bit more than individual runs within a batch, but the strategy used to score single runs may give up too many runs from the batch to be worth it.

That's certainly my take on it. Especially early in a game, you have no idea what the final score is going to be. Doing anything other than maximizing your chances to score as many runs as possible doesn't make sense.

I think there are a couple problems with what Dave has done.

First, I think park factors are very important. The Dodgers win more games than "expected" ... but that's in comparison to a run distribution which doesn't fit the Dodgers context. In low-scoring Dodgers stadium, everybody is going to have "above-typical" peaks of "low-scoring" games.

Same goes with his Rockies example. If the 2000 Rox had a "typical" run distribution, they'd have gotten murdered.

In short, the win probabilities of a given number of runs scored change by the run environment, which changes by park.

It would be interesting if Dave re-ran these using only road games or only for games played in fairly "normal" stadiums.

The second problem is more one of presentation than results. Dave underplays the importance of mean differences, giving the impression that he's looking at variance, not the mean. But he's not really. The mean is an important part of any team's distribution. The 2000 White Sox may or may not win more games than expected given their run distribution, but the fact that they scored 4 or more runs 70% of the time (i.e. they had a higher mean RS) means that, relative to a typical distribution, they're gonna win a lot more games ... certainly a lot more games than the 2003 Dodgers (relative to that same typical distribution).

Differences in the variance, given a certain mean, might add a couple games here or there. But differences in the mean will have much larger impacts. The 2004 Cards outscored the NL mean by .64 runs per game, or about 100 runs, or about 10 wins.

The 2005 White Sox run distribution might (partly) explain why they're 4 games above their pythag, but they've got a pythag of .621 because their mean RS is much higher than their mean RA. In their specific case, it's mostly their low mean RA ... which gives their opponents above-average peaks in low-scoring games which doesn't seem to be doing their opponents any good (despite the 2000 Dodgers example).

Thanks, Walt. I agree about the park factors. But I didn't want to make a complicated subject more complicated. I was careful to use my estimates as estimates, and not "true" pythagorean projections.

But I don't understand your second point. I was careful to not use phrases like variance, and to just talk about distribution. I agree that distribution includes both mean and variance (and probably one or two other things I'm not aware of). And I never mentioned that the Sox's pythagorean variance was totally due to the distribution of runs scored. I've already mentioned the Sox's low RA per game in a previous column.

I thought it was "Studemund"

I just went with the most common spelling of my last name. Right or wrong!

Either way it clearly comes off as something "made up".

If I were you I'd try to think of something shorter and snappier.

The 2004 Cards outscored the NL mean by .64 runs per game, or about 100 runs, or about 10 wins.

Which given their run distribution maybe translates to "only" 8 wins. The point is that differences of means of this magnitude happen every year while the biggest "distribution" gap of the last 5 years is 5.5 wins by the Dodgers which (1) compares them to the wrong distribution and (2) Fret suggests it's only 3.3 using the "better" pythag.

This of course has obvious implications for strategy. Employing a strategy (say early bunts) that gives you a "better" distribution but a lower mean is not likely to pay off. (whether it does or not depends on the specific numbers involved, including, most importantly, the distribution of your RA)

In other words, maybe the 2005 Sox are employing stategies that give them a better distribution GIVEN their average of 4.84 runs per game. But if those strategies are reducing their mean scoring from 5.3 to 4.84 runs per game, that's likely at best an even trade.

Or to put it another way, Dave compares the 2003 Dodgers to what you would expect (based on pythag apparently) from a team with that same mean scoring. But compare the 2003 Dodgers to the "average" team's run distribution, and they ain't 5.5 games "better".

This is another place where Dave's article is unintentionally misleading. In the graph, he shows the Dodgers distribution vs. the "typical" distribution. But (and I quote, emphasis added):

the offense that contributed the most wins over the average distribution was that of the 2003 Dodgers' team, about 5.5 wins more than you would expect given their average number of runs scored.

They did not have "the most wins over the average distribution", they had the most _given their mean RS_. Using approximations from the graph and the expected win% for each RS, the Dodgers of course underperformed the average distribution -- by about 16.5 wins over 162 games. In 53% of their games, they scored 3 or fewer runs and would be expected to win just 15 of them. Hard to recover from 15-71, even with 5.5 extra wins.

What this means is that a team with the Dodgers' RS distribution playing in an average park would underperform by about 16.5 wins relative to an average distribution. Granted, that's better than 22 wins worse. But it's clear that the mean is a hell of a lot more important than the distribution.

I think run distribution might be better to explain a team that's doing well despite having allowed more runs than they've scored. I've just run the figures on the Nationals, and the data looks like this:

Runs Times Scored
04
18
28
316
410
512
63
75
82
94
100
113

(Note: I know that's going to come out wrong, if someone can help with the formatting.)

The interesting part is the spike at 5, especially given the Nats' record in those games, 10-2, as opposed to 4-6 at 4 runs, and 8-8 at 3 runs. They also have not lost a game in which they've scored more than 6 runs, which is probably more a sample factor than anything.

It seems as though this is a case where run distribution plays out better, since there's no really high data point (more than 11) to skew the data.

But it's clear that the mean is a hell of a lot more important than the distribution.

Don't disagree. And good point about the wording on the Dodger's average and distribution. That was clumsy of me!

But I do think that the article is misleading based on what you're reading into it, not what I put there. Nowhere, for instance, do I mention the use or effectiveness of one-run strategies. I just made the point that sometimes distribution of runs scored can have an impact beyond what averages capture.

I plan to talk more about the implications of this in my next article.

I agree about the park factors. But I didn't want to make a complicated subject more complicated. I was careful to use my estimates as estimates, and not "true" pythagorean projections.

Fair enough. But when the "best" and "worst" performers turn out to be a team in the extreme pitchers park and a team in the extreme hitters park, I think you have to bring it up.

But I don't understand your second point.

I explained it more fully in my second post. The statement that the Dodgers beat the average distribution by 5.5 wins is really misleading (surely unintentional).

As I said, that second point wasn't about the results, it was about the presentation. In the graph, you show the Dodgers distro vs. the average distro (fine) but when it came to calculating their wins, you no longer compared them to the average distribution.

If you're clearer, I think your results will be misinterpreted less frequently. Just the other day, following your first piece, we saw some folks suggesting bunting in the first inning would be a good idea if it maximized your chances of scoring 3-5 runs. But if frequent bunting decreases your mean RS, this will most likely hurt you more than lowering your variance will help you.

Lowering the mean and variance of your scoring will help get rid of some of those "extraneous" runs. But it will also get rid of some of those very vital 3rd, 4th, and 5th runs. I think it's easy to view your presentation as suggesting that's not the case.

Or as I suggested above, the Sox are not winning so much because of their run distribution but because they're holding their opponents run distribution to something that looks more like the 2003 Dodgers.

By the way, the "worst" 2000 Rox outperform the average distribution by about 12 wins.

By the way, the "worst" 2000 Rox outperform the average distribution by about 12 wins.

Well, I applied park factors, see...

Kidding! Good point about park factors. I'll add a comment in the references section of the article.

Either way it clearly comes off as something "made up".

There are 25 addresses in the US with Studes' correct spelling of last name (not how Dial spelled it). There are 90 with "Studeman". And, there actually is a Dave Studeman. I like to use The Ultimates for directory assistance.

Thanks, Tango. I would have guessed that, given all the variations I've seen. I hope the real Dave Studeman doesn't come after me!

This may be obvious (but indulge a newbie)... But it seems to me that the problems with the pythag method noted here stem from the fact that the mean doesn't represent the distribution that well (and maybe that what's being argued, I'm not sure)... For instance, if the median was used rather than the mean for RS and RA (and using an exponent of 2, for simplicity), I find a median-based pythag WP of .694 for the White Sox (through 74 games), as compared to .619 from the mean-based pythag... The actual WP is .676...

I did a quick spin through the 2004 AL numbers provided in the speadsheet linked above... The overall result (from this limited set of data) is that the median works better in some cases, but is worse overall due to a few cases (the A's are the worst since median RS < median RA, due to a strange RA distribution, but their actual WP was > .500)...

I think my point is that if one could find a "perfect" number to represent the distribution then the pythag method would work better (more consistently?)... I don't, however, believe that such a number exists for all distributions because of all the variation from team to team and RS to RA... Not to mention the inordinate amount of extra work...

I think I've just reasoned myself to the simple conclusion that when the pythag method breaks down you should (or could) look at the distributions to gain some insight, which is probably why the article was written in the first place...

Er, the "made up" part was supposed to be a joke.

Sorry for the thread hijack, but since you're going to read this - and I'm going to throw in a sincere bit of flattery (I love the THT stats section and I buy the book pretty much just to keep that stuff free ) - let me ask you a question.

Are the FIPs reported at THT park adjusted?

No, FIPs aren't ballpark-adjusted at this time. However, we'll be incorporating "expected FIP" into the stat reports soon, and that will adjust the home run portion of FIP by outfield flies allowed and average home runs allowed in a given ballpark.

The issue is that we're trying to have a number that's directly comparable to actual ERA, so if we adjust FIP for ballpark (which I guess would just be on the home run and add-on portions) we should probably park-adjust ERA too.

Thanks for the compliment, Philly. Hopefully more people will buy the books to keep the stats free. That's the goal.

Either way it clearly comes off as something "made up".

If I were you I'd try to think of something shorter and snappier.

I would suggest Dirk Studeler

I think I've just reasoned myself to the simple conclusion that when the pythag method breaks down you should (or could) look at the distributions to gain some insight, which is probably why the article was written in the first place...

Well, I'm sure someone like Walt or Tango would put this differently, but I think that when Pythag is off, it's off for one of two reasons:

- The actual distribution of runs scored or allowed per game, as covered in the article, which might account for, at most, a quarter of the observed variance, or

- The timing of the distributions. That is, when the run scored distribution matches up against the runs allowed distribution on a game-specific basis. My guess would be that this accounts for the vast majority of variance.

One-run games (as well as two-run and three-run games) are the manifestation of this second cause. That's why they get so much attention in pythagorean variance discussions.

Thanks for the compliment, Philly. Hopefully more people will buy the books to keep the stats free. That's the goal.

Wha -- ? You mean the goal of people buying the book isn't so that we'll make untold millions?

Damn. Fooled again.

I wish that Nigerian Finance Minister with whom I entrusted all my bank account information would send me that check he promised, already.

That makes sense...

In general is the median shunned due to the extra effort (for little/no gain) required over the mean? I presume somewhere along the way someone (or ones) figured out that it wasn't worth, but like I said I'm new and haven't run across any mention of the median...

Guys, thanks. I have nothing to add but to say that it's threads like this that keep me coming back to Primer. I enjoyed the article studes, and I think I've learned a good deal from the discussion.

Oh, that MF THT book is freakin' awesome.

I love those stats.

Studes, can I ask a favor on some of those?

<I>I think my point is that if one could find a "perfect" number to represent the distributio<i>

Charlie Brown: that's what my run distribution does. You can come up with a "control value" to best-fit to each team's distribution for runs scored and runs allowed.

Studes, can I ask a favor on some of those?

Anytime, Chris. Drop me an email.

Thanks Tango... I'll take a look...

I still think the 5.5 win (or 3.3 win) figure would remain largely the same after adjusting for park. But I'm no longer sure of that.

So how would we do the park adjustment, anyway? There are two sets of figures that need to be adjusted. The first is the runs-per-game distribution of an average team in Dodger Stadium or Coors Field. I think Tango's program (or a mathematical approximation) would work for that task.

The second is the probability of winning a game given that you scored 1, 2, 3, etc. runs in a certain stadium. It wouldn't be easy to get that information from the data of actual results, due to distortions caused by the home team's offense and defense. One could approximate by the following type formula:

P(win given 3 runs scored) = P(other team scores 0-2 runs)/[1 - P(other team scores 3 runs)]

I think that would work if the park factor stayed constant for all games. But it doesn't: wind speed and direction, temperature, and the home plate umpire all affect run scoring, and these factors vary more in some parks than in others.

So, I'm stuck. Any ideas?

fret, my run distribution program also includes the win probability of two such distributions.

You set the control value such that you best-fit reality.

(It would be helpful to download and run that program to continue my point.)

To get a run distribution, you set the control value to .7667. This fits in with what happened in baseball from 1974-1990. (and pretty much throughout all of baseball history).

However, when you take two such distributions, the win probability that results does *not* conform to reality. My program assumes that these two distributions are random, but they are not. Most notably, both teams play in the same park. And, each team's style of play changes based on the score.

As a result, in order to fit to the actual win distributions, you have to change the run distribution (but not the mean) to match that. The result is to force the control value to .852 to match how two such distributions actually match up in reality.

Tango, that's a helpful description. Thanks.

What exactly is the output? To get an "answer" for w/l, do you have to run the two run/game distributions against each other?

Tango, excellent.

I have one question. Say I run the program with the control value set to .852. For my two teams I get something like

Team A
P(0 runs) = blah
P(1 run) = blah
etc

Team B
P(0 runs) = blah
P(1 run) = blah
etc

I want to know the chance that team A wins if they score, say, 3 runs. Is it appropriate to use the formula I gave in #32? Or, is the distribution set up so that another formula is more appropriate?

You got it right.

studes, if you put something like
5 4 .85 .82 1
in my program, that means you have a 5 RPG team with a control value of .85 facing a 4 RPG team with a control value of .82.

The win*.txt file gives you the win percentage of 5 v 4.

The *.html file will give you the frequency of each team scoring 0,1,2... runs per game (as well as scoring 0,1,2... runs per inning).

Think of my program as Poisson, but applicable for baseball.

Okay, after thinking this through some more, I am skeptical that this method will work. There are two reasons.

First, let's say the Dodgers score 3.5 rpg and an average team scores 4.5 rpg. Suppose the park factor of Dodger Stadium is 95, so that the Dodgers would score 3.68 rpg in an average park. Now if we run Tango's program with a 3.68 rpg team against a 4.5 rpg team, with control value = .852, it is telling us the chance that a true 3.68 rpg team would beat a true 4.5 rpg team averaged over all parks. But, we do not want to average over all parks. We want to do a weighted average where Dodger Stadium is weighted equally with the rest of the league combined. So the output will be a little off from what we want.

Second, and more importantly. If I'm not mistaken, the value of .852 was chosen to give a good answer to the above question: what is the chance that a team with a true rpg of x beats a team with a true rpg of y, averaged over all parks? But we are trying to use the program for another purpose. We want to know: what is the chance that team A scores exactly m runs, and team B scores exactly n runs, given that we know the park (but not the wind speed, etc), for all values of m and n?

Here is my main point. The program gives an answer to our second question for each pair m,n. But it was designed for the first question. So all we know is that when we add up the values in a certain way, it gives the right answer. The values themselves could easily be off.

Anyway, I bet my complaints don't amount to much of a real difference in practice. But if the goal is an accurate park adjustment, proper methodology is important.

Nice article by this "writer."

Anyway, I bet my complaints don't amount to much of a real difference in practice. But if the goal is an accurate park adjustment, proper methodology is important.

I agree, on both counts: the methodology is imperfect, but there likely isn't a real difference in practice. The error bars on park factors are so huge that these minor differences in methodology are drowned out among noise.

I would agree that it'd be marginally better to figure out a W% at home (adjusting the league average team to LA and using the Dodgers' actual rpg), a W% for the road (doing the reverse), and combining them. But once you start doing this, you suddenly start to see lots of little adjustments you can make:

-- as long as we're separating home and road games, build the HFA into each estimated W%
-- the average of a team's road games isn't necessarily equal to a neutral park; figure the true park factor for road games.
-- better yet, use the team's real schedule to calculate their W% in each park, then combine them all
-- now that we're going to the actual schedule and going series-by-series, adjust for the level of competition.

Anyway, I guess my point is that there are always marginal improvements that can be made. In this case, I think the noise in the park factor is greater than the slight improvements that any of these adjustments would provide.

Slightly on the topic of home/road games, you might be interested in the article at the end of this BTN publication.

I agree that the differences won't amount to much. Note that I did preface my comments that because of the nonrandom nature (e.g., both teams play in the same park at the same time, and the score dictates strategies), the control value has to be bumped from .7667 to .852. You can use the former value to see how things shake out if these other variables are not in play.