Are 1-run games more common in 2012?

There have been some funky things going on with 1-run games in 2012. The Orioles have played 25 of them already and won 19, a remarkably high percentage. (This is thanks mainly to luck and unlikely to be maintained the rest of the season.) The Athletics just swept four games from the Yankees and won each one by 1 run. The Phillies had been 11-18 in 1-run games until winning two such games in a row the last 2 days.

All of this 1-run-ness has me wondering if it’s more common in 2012. Certainly, it would seem likely that 1-run games would be more common when overall scoring is lower. When teams average closer to 4 runs a game than 5, it means that a higher fraction of final game scores will be 1-run decisions. There’s also something to be said for strategy–when overall scoring is lower, managers are more likely to “play for” 1 run, i.e. use sacrifices to advance runners, lessening the chance of a big inning.

Anyway, there’s a quick way to look up the basic numbers for such a study.

Looking at Baseball-Reference.com’s 2012 league standings page, each team’s performance in 1-run games is provided. The league average is also provided. Through yesterday, the average major league team is 48-48 overall and 13-13 in 1-run games. Obviously these numbers are rounded, so let’s not put any significant digits on our forthcoming calculations. With 26 out of 96 games being decided by 1-run, that means 27% of each team’s games fit the bill so far in 2012.

Using the same data for last year, the average team comes out to 80-80 (remember the rounding) with a 25-25 record in 1-run games. That’s 31%. So, it seems that so far, 1-run games are actually down in 2012 compared to last year.

Here are calculations going further back:

2010: 81-81, 24-24, 30%
2009: 81-81, 21-21, 26%
2008: 80-80, 22-22, 28%
2007: 81-81, 22-22, 27%
2006: 80-80, 22-22, 28%
2005: 81-81, 23-23, 28%
2004: 80-80, 21-21, 26%
2003: 80-80, 21-21, 26%
2002: 80-80, 22-22, 28%
2001: 80-80, 21-21, 26%
2000: 80-80, 23-23, 29%
1999: 80-80, 22-22, 28%
1998: 81-81, 23-23, 28%
1997: 80-80, 22-22, 28%
1996: 80-80, 23-23, 29%
1995: 72-72, 19-19, 26%
1994: 57-57, 16-16, 28%
1993: 81-81, 24-24, 30%
1992: 81-81, 25-25, 31%
1991: 80-80, 24-24, 30%
1990: 80-80, 23-23, 29%
1989: 80-80, 23-23, 29%
1988: 80-80, 24-24, 30%
1987: 80-80, 23-23, 29%

There does appear to be at least a vague correlation with scoring. For the 7 years up to the beginning of the Steroids Era, the data above sums to 562-562, 166-166, or 29.5%. For the next 16 seasons (1994-2009), the data sums to 1253-1253, 343-343, 27.4%. That’s a pretty significant gap, showing that 1-run games have been lower since the offense went up in the Steroids Era.

Then, for the period comprising 2010, 2011, and 2012 year-to-date, the totals are 209-209, 62-62, 29.7%. So, as scoring has dropped the last couple of years, 1-run games have gone back up in prevalence to their pre-Steroid levels.

Nothing earth-shattering here…just makes sense.

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Kenny
Kenny
12 years ago

I live in the Philadelphia area, and I have been bugging Andy about what seemed to me to be an unusually large number of 1-run games played by the Phillies. I think that’s what spurred Andy analysis.

Ed
Ed
12 years ago

I can’t speak for the Orioles but I’m not 100% convinced that a team’s record in one-run games is based on luck. My Indians are a good example. They’re 14-6 this year in one run games. Why is that? Seems to me the answer is as follows: 1) The front of the bullpen (mostly Perez and Pestano but also Rogers and Smith) have been very good. These guys generally only pitch when the Indians have a lead and they rarely squander that lead. 2) The back of the bullpen on the other hand is really bad. Manny Acta tends to… Read more »

Ed
Ed
12 years ago
Reply to  Andy

Andy – I thought my role was to be a contrarian. 🙂

John Autin
Editor
12 years ago
Reply to  Ed

But Ed, aren’t you just sort of moving the focus of the luck? Most teams with great one-run records also got great frontline relief performances. One reason that one-run records are broadly chalked up to luck is that outstanding relief performance in one half or even one year does not necessarily reflect true ability. I’ve often praised what Pestano has done this year; but while I believe he’s GOOD, I don’t believe he’s SO GOOD that his rate of holding leads will stand up the rest of the way. Ditto Chris Perez. Ditto Jose Valverde during his “perfect” 2011; as… Read more »

Hartvig
Hartvig
12 years ago
Reply to  John Autin

” Last year’s Yankees, with Mo and Robertson having great years, were 21-24 in one-run games.”

I’m trying to figure what the explanation is for that. Ayala and Wade were also outstanding, Logan & Joba were good. Soriano came around and was OK. They had a few stiffs but nobody that pitched more than a few innings or in a couple of games.

Could their above average offensive be part of it?

It just seems so counter-intuative that you could have an outstanding bullpen and a poor record in one run games.

Ed
Ed
12 years ago
Reply to  John Autin

John – Not sure I fully understand your comment. Nevertheless… My basic point is that I don’t think records in one run games are 100% luck and that certain teams might be expected to do well in one-run games. (Bill James came to the same conclusion though his conditions were different than what I’m proposing above). So yeah, some of this is hindsight, but some of it comes from watching the Indians perform this year and watching how Manny Acta uses his bullpen. That being said, I think one of the issues with studying one runs games is that they’re… Read more »

John Autin
Editor
12 years ago
Reply to  Ed

Ed, I know I didn’t make myself clear. I’ll try to focus. 🙂 First off, as you just noted, there’s not just one flavor of one-run games. Sometimes you take a big lead into the 9th, start to cough it up, run through your best relievers, and just barely hold on with the last man in the pen getting the last out on an at-’em ball. And as Mike Matheny has said of the Cards’ poor record in one-run games, a lot of those were comebacks that fell just short. But to the extent that we talk of one-run games… Read more »

no statistician but
no statistician but
12 years ago
Reply to  John Autin

A couple of factors in one-run games not addressed here so far are 1) scoring first, and 2) keeping the lead throughout the game. The first may be a little problematical, although all shutouts are determined in this fashion, whether the score is 1-0 or 10-0, and in low scoring eras in the past, I’d guess that putting the first run on the board meant a great deal. As for keeping the lead from the early innings on, all those comebacks that fell short for the Cards make some kind of sense in this context: score early and often and… Read more »

Ed
Ed
12 years ago
Reply to  John Autin

John – No worries…I’m mostly talking off the top of my head and doing some arm chair theorizing without any proof. Just speculating. I appreciate the additional explanation and better understand where you’re coming from.

tag
tag
12 years ago
Reply to  John Autin

I think the major reason one-run games are ascribed to luck is because, as John said, being good in them does not have much predictive value. In every sport, the number of times you crush your opponent (what one of my basketball coaches used to call jugular wins) is far more indicative of how good a team you are and far better predictive of future performance than the number of close games you win. Which makes perfect sense. In a big win, you’ve closed the door on the other team decisively, for good. In a close game, whether it be… Read more »

Jim Bouldin
12 years ago
Reply to  John Autin

Whether success in one run games is explainable by “luck” or not is a question that most definitely can be addressed analytically, with a binomial probability model and some simulations.

Jim Bouldin
12 years ago
Reply to  John Autin

Just a couple quick calculations using the Orioles and Indians examples mentioned above.

Orioles:
probability of winning 19 of 25 by chance alone = .007

Indians:
p(winning 14 of 20 by chance alone) = .058

p(both events occurring due to chance alone) = .001

Jim Bouldin
12 years ago
Reply to  John Autin

Looking at all 30 teams records in 1 run games this year, the probability of getting the observed result (absolute difference in W-L of 58 games out of 423 played), by chance alone, is very similar to the probability of the combined result of the Orioles and Indians record in such games this year, .0017 to be exact, or less than 2 in 1000.

Jim Bouldin
12 years ago
Reply to  John Autin

“Given those factors, I definitely would not bet on Baltimore to continue winning anything like 3/4 of their one-run games the rest of the year. In fact, I would gladly bet that they go .500 or worse in one-run games from here on out”

I’ll take that bet. Loser has to shoot his TV just before the playoffs start.

Jim Bouldin
12 years ago
Reply to  John Autin

Wait, no that’s not fair, John may well have already shot his TV by that point, and I, well I have neither a TV (like Ed) nor a shotgun to shoot it with.

Loser has to sing, at the discretion of the winner, the fight song of either Michigan or Ohio State, at the start of the fourth quarter of said football contest, in a public venue, clad only in briefs of a color designated by the winner. Winner also has discretion of uploading said performance to YouTube at his discretion.

John Autin
Editor
12 years ago
Reply to  John Autin

@26, Jim — It’s for the very purpose of saving my (very old) TV that I have attached a sort of “boot” that prevents it from showing Mets games.

Thanks for taking the bet. Remember, when singing “The Victors,” it’s very important to clap rhythmically throughout, except of course when thrusting your right fist in the air on “HAIL!”

Since your hands will be occupied, you’ll want to make sure your briefs provide full coverage. 🙂

tag
tag
12 years ago
Reply to  John Autin

Jim, I think making sense of this issue depends on how you define “luck.” For me and most baseball fans I don’t think the definition is narrowly mathematical. For instance, about two weeks ago I watched a game in which a team, I forgot which, went ahead in like the seventh inning against the other team’s bullpen on a one-out base hit, a stolen base, a ground out that sent the runner to third and then a blown-call infield single (it was close but the replay clearly showed the call was wrong). Obviously, there was skill involved in the initial… Read more »

Jim Bouldin
12 years ago
Reply to  John Autin

tag, your points present chance for a perfect seque into what statistics–and particularly sampling error–is really all about. I agree that when games are close, the types of events you describe can turn their outcome one way or the other. But the issue gets at sampling error, a bedrock principal of much of statistical analysis. Although in any individual game the “fluke” or “chance” events you describe may certainly arbitrarily determine the winner, their influence on who wins and loses when assessed over many such trials (games in this case), diminishes in direct proportion to sample size. This effect of… Read more »

Ed
Ed
12 years ago
Reply to  John Autin

Did I miss something? How did Ohio State and Michigan get inserted into this???

Jim Bouldin
12 years ago
Reply to  John Autin

Here’s some more evidence bearing on whether the Orioles, Indians etc are actually more skilled at winning 1-run games than are other MLB teams. This is based on assessing the likelihood of team in question’s record in such games (actually, absolute departure from .500), relative to that of all other teams in MLB. Taking the Orioles first, all other MLB teams have a combined departure from .500 of 45 games out of 398 played, not counting games played against the Orioles. The Orioles have a combined departure of 13 games after 25 such games, due to their 19-6 record. The… Read more »

Ed
Ed
12 years ago
Reply to  John Autin

Jim – I’m confused on one thing. How can all teams have a 45 game departure from .500 given that every game includes a winner and a loser?

tag
tag
12 years ago
Reply to  John Autin

Jim, thanks for your cogent and comprehensible explanation about the statistics. My belief has always been that winning one-run games is probably not “purely” luck, but that it might as well be since no one (at least not yet) can isolate any strong reasons for the success. People have looked at teams’ records in one-run games over five-year periods and compared them in each year to the year subsequent and found no correlation from one year to the next. They got R squareds of less than .01 to use your terminology. An ability to play small ball and pitching, specifically… Read more »

Jim Bouldin
12 years ago
Reply to  John Autin

Ed, it’s actually a 90 game difference (as of yesterday’s BR.com data) but you have to divide the numbers by two, because each positive departure (win) is necessarily accompanied by a negative departure (i.e. each game produces two outcomes). This is to keep from artificially inflating the sample size in the analysis.

Jim Bouldin
12 years ago
Reply to  Ed

tag, the fact that a consistent set of explanatory variables hasn’t been identified to explain 1-run game success does not mean that the phenomenon is due to chance. For one thing, teams might be accomplishing these wins (or losses) by a variety of mechanisms, both from team to team and even within a single team. Such as John discussed above, ie nail-biters all the way versus big leads you almost but not quite blow. However, the variation from year to year in success at 1-run games–that *is* potentially important and interesting and indeed I’d like to look at exactly that.… Read more »

Jim Bouldin
12 years ago
Reply to  Ed

I should probably explain why I agree with Ed’s larger point that these records by the Indians and Orioles (and also Giants and Braves) are unlikely to be due to “luck” (aka “chance”, or sampling error for the statistically inclined). And why John should buy his scarlet and gray Buckeye Briefs now while the price is good. Using the Orioles’ example, as already stated, the binomial probability of winning 19 of 25 1 run games is about .007. Let’s say we’re not quite convinced that this .760 win percentage is “real”. They might actually just be a .500 team in… Read more »

Jim Bouldin
12 years ago
Reply to  Jim Bouldin

R code for the previous. Replace .470 with the other possible win percentages (i.e. .760, .64 and .50) to get the other p values. Based on 100,000 random draws from a binomial distribution.

probs = rep(NA,100000)
for (i in 1:100000) probs[i] = rbinom(1,16,.470)
length(probs[probs>=8])/100000

[1] 0.50146

John Autin
Editor
12 years ago
Reply to  Jim Bouldin

Jim — I’m suspending my trash talk for the moment. I don’t know as much probability as I should, so please forgive my layman’s terms. But: It seems to me that “winning or losing one-run games” varies from the classic probability models of coin-flips or dice-rolls in at least one hugely important way: The individual outcomes that we call “one-run wins” and “one-run losses” are subject to an enormous volume of noise, which affects not only the “win” or “lose”, but whether any single trial, once commenced, even winds up being counted. Therefore, it takes far more trials to get… Read more »

tag
tag
12 years ago
Reply to  John Autin

John, I was going to argue something along those exact same lines but, coward than I am, threw in the towel by doubting that we could isolate the variables sufficiently to know the precise reasons a team is successful at one-run games.

Anyway, I’m glad to see you’re fighting the good fight and hope you don’t end up bellowing publicly in those scarlet and gray briefs. 🙂

John Autin
Editor
12 years ago
Reply to  John Autin

P.S. In my comment #33, paragraph 4, I should not have said “noise,” but rather, “influences.”

Jim Bouldin
12 years ago
Reply to  John Autin

John, I think there’s some confusion here so it’s good that we discuss and clarify it. The original post stated that the Orioles 19-6 record in 1-run games was due to luck and unlikely to continue. At some point, somebody conflated this idea with how good of a predictor such a record is for how good a team is overall, ie in all games. But those are two very different things and we need to keep them separate. I’m discussing only how skilled versus “lucky” a team is at winning 1-run games only–I’m not arguing at all, one way or… Read more »

John Autin
Editor
12 years ago
Reply to  John Autin

@41, Jim — All my comments here have been meant to focus on the original issue — whether one-run records predict future one-run results — even when I did mention over-all records. But let’s move on. Towards the end, you wrote: “the underlying assumption of the kinds of analysis I have presented is that the systems analyzed will remain +/- constant over the full season.” That’s our fundamental disagreement. Or maybe it’s our point of misunderstanding? Maybe we’re just not taking the same meaning from the question of “is Baltimore’s one-run record due to luck and thus unlikely to continue?”… Read more »

John Autin
Editor
12 years ago
Reply to  John Autin

P.S. to my #45 — The restrictions I put on labeling the 20-sided die were actually too conservative. A better model would be labeling 4 to 8 sides, with a ratio of W’s and L’s no greater than 5:2. That would, of course, create greater uncertainty at the meaning of 97 rolls. There have been some teams that had a good deal more than 35% one-run games. The 1968 White Sox had over 45% (74 of 162), and some other teams that year were also over 35%. And there have been a few that won or lost more than 2/3… Read more »

Jim Bouldin
12 years ago
Reply to  John Autin

“That’s our fundamental disagreement. Or maybe it’s our point of misunderstanding? Maybe we’re just not taking the same meaning from the question of “is Baltimore’s one-run record due to luck and thus unlikely to continue?”” No, the comment I made (regarding the system staying +/- constant) that you are referring to there is definitely NOT the issue. If the system changes–that is to say the teams’ ability to win 1-run games changes, that doesn’t change the fact that when they were winning such games, they were not doing so by chance. Anyway, let’s start with your last sentence in that… Read more »

Jim Bouldin
12 years ago
Reply to  John Autin

“After 97 rolls, I’ve had 19 W’s and 6 L’s. What is the probability that this is a reasonably good representation of the probabilities that you fixed on the die? And what is the probability that this gives me a good prediction of the next 65 rolls?” OK right there’s the problem. Or two, because it addresses two separate issues. I realize that both Andy and you are interested in your second question there, because you’ve both stated so. Whether the Orioles or Indians will keep up this pace in winning 1-r games (or e.g. Toronto in losing them) is… Read more »

John Autin
Editor
12 years ago
Reply to  John Autin

@48, Jim — As you intimated, the different reasons why we approach the question — one for pure research, one for predictive value — inevitably color how we think about it, and particularly, just what we mean by “luck.” I grant that, using one meaning of luck, we could show that Baltimore’s 19-6 mark was NOT luck. The late-inning relievers DID slam the door when they had a small lead. Perhaps the offense DID come up with just the right size of late rally. Conceivably, at least, those things DID happen repeatedly. But in practical terms, what do we mean… Read more »

Jim Bouldin
12 years ago
Reply to  John Autin

John this is a very important conversation because it gets to what we mean by the terms chance, luck, probability and so forth. There are at least, four different ideas going on here that could be explored in great depth (and have been by statisticians). One of the issues I hear you alluding to is that of sample size. There’s no question that a larger sample size will give one more confidence in whatever statistical results are obtained from it. This is in fact why I initially gave the results for all 2012 MLB 1-r games to date–to show that… Read more »

John Autin
Editor
12 years ago
Reply to  Jim Bouldin

Forgive me for going on…. Here’s Baltimore’s one-run record since 2009: 2009 — 17-22 2010 — 29-21 2011 — 22-22 2012 — 19-6 Their Pythagorean total win projections in those years (Pythag is proven to be a better predictor than actual record) were 69, 63, 67 and 72 (projecting 2012 season total). Without significantly changing the objective quality of the team, their one-run record has wavered from bad, to good, to neutral, to fantastic. The 2010 team was worse over all than 2009, but their one-run record was far better. The Marlins: 2009 — 30-20 2010 — 23-28 2011 —… Read more »

John Autin
Editor
12 years ago
Reply to  Jim Bouldin

@28, Jim — The flaw in this particular example, as I see it, is your assumption that Baltimore’s 19-6 one-run record necessarily pegs them as objectively having AT LEAST a “.500” skill level in such games. As stated in my other recent comments, I think that assumption is completely unfounded. As far as I can tell, you plucked that .500 floor out of the air. The logic seems circular to me — you seem to imply that a 19-6 mark is significant enough to establish a .500 floor, and then use that floor to help show that a 19-6 mark… Read more »

Jim Bouldin
12 years ago
Reply to  John Autin

OK lots of confusion arising, as I was afraid might happen (since it usually does on these kinds of topics), and getting right to the heart of what statistical analysis does and does not tell us.

Hang on, this is going to require a dissertation of sorts.

John Autin
Editor
12 years ago

Good stuff, Andy.

Using the same method:
– 1908, 32%
– 1968, 35%
– 1972, 32% (AL, 34%)
– 1976, 34%

The inverse correlation between average scoring and one-run games is undeniable. Focusing on the percentages, I was at first surprised that the effect wasn’t even larger. But then I put it back into actual games — the difference in one-run games between 1968 (35%) and the more normal scoring levels of recent years (just under 30%) means an additional 8 or 9 one-run games per team in the Year of Our Gibson.

Doug
Editor
12 years ago
Reply to  John Autin

In that 1976 season, the Expos played 60 one-run games, going 27-33. Two years later, they had 59, going 23-36, a .390 clip for a team with a .519 Pythag. And, their manager (Dick Williams) was, by all accounts, a pretty shrewd tactician.

Jim Bouldin
12 years ago
Reply to  John Autin

“the Year of Our Gibson”
🙂

In the Era of the Mound Builders.

Chad
Chad
12 years ago

The Phillies just won their 3rd straight 1 run game over the Brewers – all by the identical 7-6 score.

John Autin
Editor
12 years ago
Reply to  Chad

Oh, dear…. Tough times for skippers in Milwaukee, Houston, and the never-so-aptly named Flushing, NYC.

Kenny
Kenny
12 years ago
Reply to  John Autin

Hey, don’t beat up on Flushing. I went to good old Flushing High School. The area was named by the original Dutch settlers after the town of Vlissingen in Holland.

Kenny
Kenny
12 years ago
Reply to  Andy

Don’t get your hopes up….

Jim Bouldin
12 years ago

Another illustration of the commonly observed phenomenon of variance increasing with the mean, not just in baseball but in a whole range of natural phenomena. If the variance increases, you will have fewer tight games. Why it does so however, is the really interesting question.

Jim Bouldin
12 years ago

I’m going to attempt a (somewhat) different tack on this issue to show the low probability that chance alone can explain 1-r game results. I got the records for all teams in 1-r games for the 5 years from 2007 to 2011, inclusive. I summed the absolute departure from .500 for all teams, and summed those, within each year. Then I computed the likelihood of these departures from .500, for each year: year, games, succ., p 2011, 752, 415, .0026 2010, 732, 401, .0056 2009, 656, 368, .0010 2008, 681, 367, .0181 2007, 679, 362, .0450 where succ. = observed… Read more »

John Autin
Editor
12 years ago
Reply to  Jim Bouldin

In all earnestness, Jim, I do want to follow you, but I’m lost. I’m sure your method is common in your field, but it’s alien to me. Can you break it down, step by step? And I need definitions: — “Absolute departure from .500”: Is that the absolute value of their (1-r wins minus 1-r losses)? And after summing those, what did you do with them? What’s their role in the table? — “observed # of successes” — What is that? I have a very linear mind, alas. And a completely separate point to consider: You suggest that the Orioles’… Read more »

Jim Bouldin
12 years ago
Reply to  John Autin

John, OK, I’ll try to explain it. Games is the number of 1 run (1-r) games played in that year. For each team, I get their record in 1-r games for that year, and compute the absolute difference between the W’s and L’s (so, yes to your question on that). You have to take abs values because these values are then summed over all teams (and which otherwise would always sum to a .500 record, which is meaningless). This number then has to be divided by two because there’s an L for every W, as mentioned to Ed above, giving… Read more »

John Autin
Editor
12 years ago
Reply to  Jim Bouldin

Let me try to follow along with the simplest example: A league of 2 teams, one is 20-10 in 1-r games, the other is (necessarily) 10-20. Now, my understanding of what you’ve said is that the absolute departure from .500 is 10 for each team — 20 wins minus 10 losses for Team A, and abs(10 wins minus 20 losses) for Team B. The sum of the two absolute departures is 20; divide by 2 and we have 10 again. And that’s d; d=10. Expectation is half the number of games played. Each team has 30 games, the total is… Read more »

Jim Bouldin
12 years ago
Reply to  John Autin

Excellent example and you almost got it. The number of games already accounts for the fact that there are two teams in each game, so games = 30, and expectation is half that, or 15. So, you “expect”, by maximum likelihood principles, for each team to go 15-15 in those 30 games, under the hypothesis that 1-r game outcomes between them are due to chance alone. You’re essentially done at this point; all you have to is apply the binomial probabilities, which is always determined as: p(successes, trials, p(i)) which is read as: the probability of obtaining the observed number… Read more »

Jim Bouldin
12 years ago
Reply to  John Autin

And one last, but important, point and then I’m going to go watch Justin mow down the Indians. One of the beauties of this is that there are only two alternative explanations for the observed results (chance versus a skill difference). Therefore, if we compute the probability p for one of them, the other one must, by definition be 1-p. From that we can easily compute the relative strengths of evidence of the two, as a ratio. In this case, p = .02 and so 1-p = .98. Their ratio is 49; so it’s 49 times as likely that the… Read more »

John Autin
Editor
12 years ago
Reply to  John Autin

@58, Jim — Thanks, but try to bring me all the way home. Two things remain unresolved in my head: (1) Did I have the right answer for d? d=10? Somehow, that doesn’t seem right. If games=30, then expectation=15, and succ.=25. Seems like succ. should be 20. I know expectation has to be 15, so I must not have used the right method to find d. (2) Once you’ve settled the “d” question, could you step out the binomial probabilities equation? Or at least draw the outline a little deeper? I’m good with algebra and arithmetic; I just don’t know… Read more »

Ed
Ed
12 years ago
Reply to  John Autin

Sorry Jim but Verlander’s struggles at Progressive Park continue!

Jim Bouldin
12 years ago
Reply to  John Autin

My fault on that John, not yours. Starting over with a (hopefully) more simplified explanation: 1. games = total number of games played. 2. d = sum of the absolute value of the deviations from a .500 record, over all teams. [e.g. for a single team that goes 18-24 in 1-r games, its d value is 6] 3. Expected wins (E) = games/2. 4. d = d/4 (i.e. divide d by 4). 5. succ = E + d. 6. trials = games so then, for your example: games = 30 E = 30/2 = 15 d = (10 + 10)/4… Read more »

John Autin
Editor
12 years ago
Reply to  John Autin

@63, Jim — Many thanks for the crystal-clear breakdown.

Is “pbinom” an Excel function, or do you know an equivalent Excel function? I’d like to be able to run pbinom(20, 30, 0.5) and get the solution 0.0214.

Jim Bouldin
12 years ago
Reply to  John Autin

“…could you step out the binomial probabilities equation? Or at least draw the outline a little deeper?”

It’s a cumulative probability distribution = the summation of all binomial probabilities for the set of possible successes >= the number actually observed. So for your example:

successes observed (succ) = 20
S = set of possible successes >= succ
= 20:30
pbinom = summation(p(S))
= p(20 wins) + p (21 wins)…+ p(30 wins)

with the binomial probability for each possible number of successes given by:
p(i)^succ * (1 – p(i))^(trials – succ)

(^ = exponentiation)

Jim Bouldin
12 years ago
Reply to  John Autin

John, I stopped using Excel years ago but it can very likely compute binomial probs. Look in the help menu. OpenOffice Calc can do it if Excel doesn’t.

The way to go on this however, without any question, is R. Very likely the most powerful statistical analysis tool known to man. And open source.

Jim Bouldin
12 years ago
Reply to  John Autin

R:
http://www.r-project.org/

Highly recommend installing and starting to explore it, you won’t believe the power of this thing.

Jim Bouldin
12 years ago
Reply to  John Autin

Andy, Thanks for pointing that out, and you are correct. However, to really determine the presence of non-independence in the data (it can’t be assumed, has to be demonstrated), one would have to compute the first order autocorrelation function (which essentially measures how strongly wins and losses in 1-r games are clustered in time). If that value were low, no adjustments to the probabilities would be needed. If that value were high however, then yes I’d have to account for it, especially at the smaller sample sizes (like individual teams within years for example). But for the composited data (all… Read more »

Jim Bouldin
12 years ago
Reply to  John Autin

Anyway, just to try to clarify the mud pool I’ve almost certainly created by now, and for those still interested and familiar with binomial prob: In the table above the p value for each year is derived as (using R lingo): pbinom (succ., games, 0.5), that is the probability of obtaining the observed number of successes in the defined number of games, given that the likelihood of success in each game is 0.5 (i.e. random chance). So for example, for 2011, a p of .0026 is obtained by the binomial probability of 415 successes in 752 attempts, e.g. 415 heads… Read more »

e pluribus munu
e pluribus munu
12 years ago
Reply to  Jim Bouldin

You’re a good teacher, Jim. You unpacked the formulas very clearly and I was surprised to understand (I think) your point and to see its validity. But I’m not sure you’ve answered Andy’s point fully. I don’t think anyone would actually argue that 1-r results are generally the result of luck. If the 1927 Yankees went 6-0 in 1-r games vs. the Browns, no one would attribute that to luck – unless they thought luck had let the Browns hang in there so many times. Wouldn’t the real issue of “luck” (or, conversely, meaningfulness) be related to deviation from the… Read more »

Jim Bouldin
12 years ago

Thanks epm, I’m glad some of this was helpful to somebody.

I’m not sure which of Andy’s points you are referring to, but on the issue of Pythagorean estimates and “luck”, I don’t think much of the underlying ideas of that concept, especially relative to success in 1-r games. It’s designed to estimate what “should” have happened (in terms of W-L record) over all games, and by subtraction, how “lucky” or unlucky a team was.

And some people have definitely argued that 1-r game outcomes are determined largely, or at least often, by “luck”.

Jim Bouldin
12 years ago

My analyses are all wrong. Well almost all anyway. I’m going to have to redo all of it when I get some time.