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Category: Fun

Two cannibals are eating a clown and one says to the other, “Does this taste funny to you?”

April 28, 2008 | 20 Comments

Hitting or Pitching. Which wins more games?

By Tim Murray and William Briggs

You obviously need to score runs to win baseball games, and buying better hitters does this for a team. But you also need to keep your opponent from scoring too many runs, and buying better pitchers does this. Good, error-free, fielding, all other things being equal, will also help a team keep the runs scored against it low. Most teams cannot afford to buy both the best batters and the best hurlers, so they have to make decisions.

You’re the newly appointed manager for your favorite team. The roster is nearly made out, and you find you have money for one more player. You can buy a hitter to improve your team’s overall batting average (BA) or you can acquire a pitcher to lower your team’s earned run average (ERA). What do you do?

We decided to try and answer this question by looking at the complete data from the 2001 to the 2007 seasons for all teams in Major League Baseball. For each team, the number of regular season Wins, batting average, earned run average, number of errors, which league American or National, and total payroll were collected. We also counted the total runs scored for and allowed for each team, but since these statistics were so closely connected with batting average and earned run average, we don’t consider them further.

Payroll is obviously used to buy what teams consider, but as fans know to their grief do not always work out to be, the best players. If winning more games was simply a matter of increasing the payroll, the New York Yankees would win every World Series. Thankfully, then, money isn’t everything.

But it is something. This picture shows the payroll by the number of wins, with each team receiving its own color (since this is for seven years, each team appears seven times on this, and all other, plots). The team to the far right in blue are the Yankees, far exceeding any other team in money spent. The club next to them in red are the Boston Red Sox. There is a huge difference in the amount of money spent between teams. The 2006 Florida Marlins spent the least at about $15 million but won a respectable 78 games. They were followed closely by Tampa Bay, which in 2000 spent about $20 million, only rising to $24 million by 2007. Their wins were steady at around 66.

wins by payroll

A horizontal line has been drawn in at 90 games to show that there is still an enormous range of team payrolls for clubs winning at least this impressive number of games. For example, the 2001 Oakland A’s spent only about $34 million to capture 102 games. They increased the payroll a mere $6 million the next year and won 103 games. Oakland, as documented in the book Moneyball by Michael Lewis, didn’t really drop much below 90 games until last year, winning only 76 games while spending the most they ever had, nearly $80 million.

While spending a lot does not guarantee winning the most games in any year, it does help. The Yankees, for example, never dropped below 94 games (in 2007). Boston was the second biggest spender, and it has helped them win at least 82 games a year. However, most teams cannot spend nearly as much these two. Other teams must be grateful that money isn’t everything.

This second picture explains why money can’t necessarily buy happiness. Each of the three predictive statistics, BA, ERA, and Errors, is plotted against Payroll. A statistical (“nonparametric”) regression line is drawn on each to give a rough, semi-qualitative idea of the relationship of the variables. The signals go in the expected direction: larger payrolls mean, on average, higher BAs, lower ERAs, and lower numbers of Errors. But none of the signals are very strong.

wins by BA, ERA, and Errors

To explain what we mean by that, pick any level of payroll, say $100 million. Then look at the scatter around that number (the points below and above the solid line). With BA, the scatter is just about as wide as the range of team batting averages in the data, which are .240 to .292. The same is true for both ERAs and Errors. Still, there is a general weak trend: spending more money does, very crudely, buy you a better team.

But not much better. For example, if you wanted to spend enough to be 90% sure of upping your team’s batting average 5 points (from the median of .268 to .273), you’d have to shell out an extra $50 million (this is after controlling for League, Errors, and team ERA). That’s a huge increase in team salaries. Even worse, the players you buy would have to have extraordinarily high batting averages to bring the entire team’s average 5 points higher. It’s the same story for ERA and Errors. The point being, is that predicting what players will do, paying more money for those you consider better, and their actual performance after you buy them is not just a tricky business, but an almost impossible one.

This still doesn’t answer what is better, in the sense of predicting more wins: hitting or pitching. Take a look at this picture:

BA, ERA, and Errors frequency by League

This shows fancy, souped-up, “histograms” (called density estimates) for the frequency of BA, ERA, and Errors by League. Higher areas on the graph, like a regular histogram, mean that number is more likely. For example, the most likely value of ERA for teams in the National League is just over 4.0.

It’s clear from these pictures that the American League teams have on average higher ERAs and BAs than do clubs in the National League. Obviously, the designated hitter rule for the American League accounts for most, if not all, of this difference. There doesn’t seem to be any real differences in Errors between the two Leagues, which makes sense. The League differences between ERA and BA have to be accounted for when answering our main question.

This next series of pictures shows there is even more complexity. The first is a plot, separated by League, of each teams’ BA by ERA. There is some weak evidence that as ERA increases, BA drops, especially in the American (A) League, perhaps another remnant of the designated hitter effect. But this isn’t a very strong indicator.

BA by ERA by League

This next pictures shows some stronger relationships. The top two panels, again separate by League, are plots of ERA (on the vertical axis) by Errors (on the horizontal axis): as ERA increases, so do numbers of Errors. Similarly for BA, as numbers of Errors increases, the batting averages of teams tend to decrease. All this evidence means that when a team is bad, it tends to be bad in all three dimensions, and when it is good, it tends to be good in all three dimensions. This is no surprise, of course, but we do have to control for these factors when answering our question.

BA, ERA, by Errors by League

We finally come to our main question, which we answer with a complicated statistical model, one which accounts for all the evidence we have so far demonstrated. The type of model we use accounts for the fact that the number of Wins is a discrete number, by which we mean the total Wins can be 97 or 98, say, but they cannot be 97.4. In technical terms, it is called a quasi-Poisson generalized linear model, a fancy phrase that means that the model is very like a linear regression model, about which you may have heard, but with some twists and extra knobs that allow us to control for our interacting factors and discrete response.

The answer lies in these complicated-looking pictures. Let’s work through them slowly. First, only look at the top picture, which is the modeled, or predicted number of wins by various batting averages.

Predicted wins

There are two sets of three curves. The brownish is for the National League, and the blueish for the American. Now, in order to predict how many wins a team will have, we have to supply four things: their expected BA, ERA, number of errors, and League. That’s a lot of different numbers, so to simplify somewhat, we will fix the number of Errors at the median observed figure, which is 104. (Changing Errors barely changes the results.)

We still have to plug in a BA, ERA, and League in order to predict the number of wins. We first start by plugging in the BA over the range of observed values, but we still have to supply an ERA. In fact, we supply three different ERAs: the observed median, and first and third quartiles, which are: 4.04, 4.37, and 4.74. For the American League, these are the three blue curves: the top one corresponds to the lowest ERA of 4.04, the middle for the value of 4.37, and the bottom for the highest value of 4.74. To be clear: each point on these curves is the result of four variables: a BA, an ERA, a number of Errors, and a League. From these four variables, we predict the number of wins, which varies as the four variables do.

All of these curves sweep upwards, implying the obvious: higher BAs lead to more predicted Wins, regardless of ERA or League. At the lowest BAs, differences in ERA are the largest in the American League. Meaning that, if your team is hitting very poorly, small variations in pitching account for large changes in the number of games won. To make sure you see this, focus on the very left-most points of the graph, where the BAs are the smallest. Then look at the three blue curves (American League): the three left-most points on the blue curve are widely separated. Moving from a team ERA of 4.74 to 4.04 increases the number of games won from 61 to 78, or 17 more a season, which is of course a lot. But when a team is batting well, while differences in ERA are still important, they are not as influential. These are the right-most blue points on the figure: notice how at the largest BAs, the three curves (again representing different ERAs) are very close together. If a team in the American League is batting very well, improvements in pitching do not account for very many more games won.

That is so for the American League, but perhaps surprisingly not for the National, where the opposite occurs. Differences in ERA are more important for high batting averages, but not as important for low ones: better pitching becomes more crucial as the team bats better. The brown curves spread out more for high BAs, and are tighter at low BAs.

Now let’s look at the bottom picture. This is the same sort of thing, but for the range of ERAs are three fixed levels of BA: .259, .266, and .272. The top curves are the highest BA, and the bottom curves the lowest. Looking first at the American League, we can see that when the team ERA is low, differences in BA do not account for much. In fact, when the team ERAs are the lowest, improvements in batting in the American League are almost not different at all! When team ERAs are high, changes in BA mean larger differences in numbers of games won: the spread between the blue lines increases as ERA increases.

Again, the situation is opposite for the National League: when the team ERA is low, changes in BA are more important than when teams ERAs are high. In this league, when team ERAs are low, good batting can make a big difference in numbers of games won. But when ERAs are high, improvements in batting do not change the number of games won very much.

Once more, we point out that we can draw each of these three curves again for different numbers of Errors. We did so, but found that the differences between those curves and the ones we displayed were minimal, but not negligible: for example, adding a whopping 40 errors onto a team that ordinarily only commits 80, on average only costs them 2 games a season. Higher BAs or ERAs can mitigate this somewhat, from losing 2 games to only losing about 1 extra game a season. So while Errors are important, they are by far decisive factors in an overall season.

So what should you do?

Look again at the two plots. In the BA plot, the highest number of predicted wins, for a BA of .292 for the ERA of 4.04 (the lowest pictured) is about 104 games for National League teams, and about 100 for American League clubs. But the hitoghest number of predicted wins, looking at the ERA plot, for teams with the lowest ERA of 3.13 with the BAs of .272 (the highest pictured) is about 111 games for the National League and 107 games for the American. Conversely, back in the BA plot, those teams with the lowest BAs of .240 and high ERAs of 4.74 won only about 61 games in the American League and 67 in the National. While—in the ERA plot—teams with the worst ERAs of 5.71 and lowest BAs of .259 won only about 56 games in the American and 62 in the National.

Clearly, then, pitching is more important than batting overall: more games on average will be won by those clubs who have the lower ERAs than those teams with the higher BAs.

But that isn’t necessarily the answer to our question. Remember that you only have money for one more player. Should you recruit or trade for a better pitcher or batter? It depends on what kind of team you have now. Our team right now has a certain ERA, BA, and expected number of Errors, so what do we do? The final answer is in this last picture.

Effects of ERA and BA

This shows improvement, in either ERA (decreasing) or BA (increasing) on the bottom axis. The other axis shows for each “unit” of improvement (0.05 for ERA, 0.001 for BA), the additional games won. These are the same, in essence, of the plots above, but they show the data in a different fashion (the same colors still represent the two leagues). The way this figure works is that you pick a certain point, say a BA of .266 or an ERA of 4.34 (which is the same point on the graph), and then move upwards (to the right on the horizontal axis) by one “unit” (0.05 for ERA, 0.001 for BA) and then pick off the number of additional games won.

No matter where we are on the graph, ERA easily wins this race, in the sense that buying a better picture to improve the ERA wins more games than buying a better batter to improve the BA. This is true for either league. (These pictures are also concocted using the median values of ERA, BA, and Error, as mentioned above: do not worry if you don’t understand this; the results do not change for the other values.)

So spend your money on the pitcher.

Tim Murray is a student at Central Michigan University and can be reached at murra1td@cmich.edu. William Briggs is a statistician in New York City and can be reached at matt@wmbriggs.com.
April 24, 2008 | 11 Comments

CONTEST: Preliminary Discussion of the “Best Internet Conspiracy Theory”

Best Internet Conspiracy Theory
This is the first posting preliminary to the announcement of an Official Contest to find the Best Internet Conspiracy Theory.

The Contest will be officially announced in about one week.

This contest is primarily a public service for those who contribute regularly to sites like Digg.com, Reddit.com, Wikipedia.org, etc. Many of those people are forced to spend an inordinate amount of time concocting theories that neatly explain messy world events. This has led to an enormous increase in carpal tunnel and internet addition syndrome cases worldwide. Thus, we want to provide these overworked souls a handful of ready-made theories to which they can refer. The theories we have in mind are described in the contest rules below.

I will need help in publicizing this Contest, and may need help in judging entries, depending on how many I receive. Volunteers should email me: put “CONTEST” in the subject line.

A sketch of the rules is as follows:

(1) All entries must be shorter than 150 words. Shorter entries will receive more weight than longer ones.

(2) Entries—one per person—must be placed into the Comments Section of the Official Contest Post. No discussion will be allowed on that post; only Contest entries are allowed.

(3) All entries will be judged by the intrinsic awfulness, brevity, completeness of derangement, plausibility, specificity (names named), and potential appeal to the everyday, e.g., Digg reader.

(4) The Contest will last approximately two to three weeks.

(5) A prize, or prizes, to be decided later, will be announced.

(6) An example of an Internet Conspiracy Theory:

Certain scientists discovered a formula, derived from an alien artifact dug up in Area 51, for turning ordinary sea water into limitless, cheap fuel. Green Energies, a subsidiary of MoveOn.org, based in the World Trade Center was about to sell this discovery and eliminate Global Warming, when the Oil Companies learned of it. Big Oil contacted George Bush, who ordered the Twin Towers destroyed before the secret could get out. Ron Paul found out about this and was going to expose the entire matter had he won the Republican Nomination, which he would have done except the Mainstream Media ignored him.

Please do NOT post any conspiracy theories now! Save them for the Contest.

April 20, 2008 | 8 Comments

It was bound to happen

Remember how you used to cavalierly ignore those “Keep of the Grass Signs” in your un-enlightened youth?

Well, you brutal, uncaring, beast.

For it has finally been announced—from Europe, naturally, from the Swiss government-appointed Federal Ethics Committee on Non-Human Biotechnology—that plants have feelings too.

They have authoritatively stated that “interfering with plants without a valid reason as ‘morally inadmissible.'” This means the next time you carve you and your sweetheart’s name into a tree can lead to a nice, long jail sentence. (If the famed Swiss police ever catch you, that is.)

The ethics committee did grudgingly admit—for now—that “all action involving plants for the preservation of the human race was morally justified.” Meaning, I suppose, that it’s still OK to eat them. I probably don’t need to explain to you the fix we’d be in if we could not. But there is only direction for the Enlightened to go, so stay tuned for an announcement banning the use of “higher” plants, such as maybe corn and tomatoes, for use in the “preservation of the human race.”

The august Swiss body has also found that “genetic modification of a plant did not contradict the idea of its ‘dignity’.” Yes, I can see how a kumquat would not find it an affront to be genetically probed. Until, that is, the kumquat learns how easily this sort of thing can sully one’s reputation. It’s only matter of time before a lawyer figures this out and brings a case to Brussels.

Just keep all this in mind, think about what you are doing—raise your awareness!—next time you are at the salad bar.

March 31, 2008 | 7 Comments

Tall men in planes

I am off to Spain today, for the conference, to present my unfinished, and unfinishable, talk. Why unfinishable? I am asking people to supply estimates for certain probabilities (see the previous post), on which there will never be agreement, nor will these estimates cease changing through time. I am somewhat disheartened by this, and would like to say something more concrete, but I am committed. So. It’s eight hours there and back, crammed into a seat made for, let us say, those of a more diminutive stature than I. There will be no more postings until Saturday, when I return, which is why I leave you with this classic column I wrote several years ago, but which is just as relevant today.
Burden of the very tall

Lamentations of the Very Tall

An alternate title of this article could have been, “Short People Rejoice!” for it’s my conviction that the world is mercilessly biased in favor of tiny people. That is, probably you.

I say “probably you” because of the firm statistical grounding in the fact that it is quantifiably improbable for a random person to be tall. I’m also assuming that you, dear reader, are a random person, and therefore most likely belong to the endless, but shallow, sea of short people.

Here’s the thing: since you are probably short you are likely to be unaware of how tall people suffer, so I’m going to tell you. For reference, I am a shade over six-two, which is tall, but not professional basketball player tall. This is still taller than more than nine-tenths of the American population, however.

Life as a tall man is not all bad. It’s true I’ve developed strong forearms from beating off adoring females who lust after my tallness, but there are many more misfortunes that outweigh the unending adulation of women. Showers for one.

Shower heads come to mid-chest on me. I’ve developed a permanent stoop from years of bending over to wash my hair—and then from scrunching down to see my reflection in the mirror, typically placed navel high, so that I can comb it.

The lamentations of the tall when it comes to airplane seats are too obvious to mention. As is our inability to fit into any bathtub or fully on any bed.

I once worked in a building that required, for security reasons, a peephole to be drilled into the door. I stood guard over two workers who dickered over where to place the pencil mark that would indicate where they were going to drill. Each in turn stepped up the door and put a dot in the spot where their eye met the door. The marks didn’t quite match but they soon settled on the difference.

Ultimately, the hole was about crotch high on me. To be fair, I was in Japan and the workers were Japanese, and therefore on the not tall side of the scale. Because I was in the military, I wasn’t entirely comfortable bending down to that degree1. This meant that I breached security each time I opened the door because I couldn’t see who was on the other side. Suspicious, is it not?

It was at this point that I began to believe that this discrepancy in height was not entirely genetic and that sinister motives may be behind the prejudices of the non-tall.

For example, I have to place my computer monitor on three reams of paper so that it approaches eye level, and I have to raise my chair to its maximum so that my knees aren’t in my chin, but when I do my legs won’t fit under the desk. No matter how I position myself I am in pain. I sit2 in a factory made cubicle-ette which, as far as I can tell, causes no difficulties for my more diminutive co-workers. This is more evidence of the extent of the conspiracy of the non-tall.

Shopping is suspiciously dreadful too. Short people can freely walk into any department store and grab something, anything, off the rack, while we tall men are stuck with places like Ed’s Big and Tall. These stores are fine if you have a waist of at least 46 inches and you have stumpy legs, but they are nearly useless otherwise.

Pants for the tall are a cruel joke. Even if they carry labels that promise lengths of 35 or more inches, we know that these labels are a lie. Yes, the legging material may stretch for yards and yards, but there is never enough space where it counts. These pants are called “short-rise” for obvious reasons. I asked a salesguy (a non-tall man, of course), do they make long-rise pants anymore? He didn’t stop laughing. Normally, I’d have my revenge by not buying anything from him, but I couldn’t buy anything from him in the first place. I could do nothing but fume.

I’m not sure how we, the tall, will be able to overcome these horrific adversities. In raw numbers we are but a small minority—a fairly imposing looking minority it’s true—but a minority just the same. Still, there is word that something can be done and I hear that we’re to discuss ideas at our next official Tall Man Meeting. Don’t bother trying to sneak in, though, because we take measurements at the door.

1If I would have been in the Navy, I would have been used to it, of course.
2This was true then; it no longer is. I do not have a desk now.