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*Baseball team and player performance examined realistically and accurately.*

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“;‘Where shall I begin, please, your Majesty?’ he asked.

‘Begin at the beginning,’ the King said, gravely, ‘and go on till you come to the end: then stop.’”

–*Alice’s Adventures in Wonderland*,

Lewis Carroll

‘Begin at the beginning,’ the King said, gravely, ‘and go on till you come to the end: then stop.’”

–

Lewis Carroll

Quick page jumps:

It gets called a lot of thing: analysis, analytics, sabermetrics, and (by some traditionalists) a number of unprintable names. But a lot of things that *analysts*—most of them, anyway—would not call “analysis” are shoved into that category by the less-well-informed (including a lot of people professionally associated with the game) just because they involve numbers. Let’s clarify that.

Things like pitchers’ pitch spin rates, batters batted-ball “launch angles”, and fielders’ positioning (“shifts”) are all too commonly lumped into “analytics”. They are not. They are important, interesting, and useful things to know about, and when intelligently used can help a team to perform better. But so can strength training, aerobic exercise, good diets, and many, many other things, none of whch anyone is likely to refer to as “analytics”.

Those sorts of things wrongly lumped in with analytics are, without doubt, included simply because they deal in accumulated data, usually numeric. One could, it must be supposed, refer to acquiring and using such data to help improve player performance as “analyzing” the game, but that is really, *really* stretching the term. What such things are is coaching tools. Using such data is a wise and honorable thing to do, but it isn’t *analytics*. So…what *is* real analytics then? We’re glad you asked.

At bottom, it is the application, grounded in demonstrable success, of ways of applying stochastic mathematics to numerical player and team statistics to see how the various elements of the game combine to generate runs and wins. Nowadays, complicated measures with cryptic names (and, often, little explanation of their exact mechanics) have proliferated, to the confusion of those not already well grounded in these arts. It is our contention that the basic ideas, and even the basic “formulae”, are not at all difficult to grasp, and we will try to demonstrate that on this page.

(Stochastic: from Oxford Languages, “randomly determined; having a random probability distribution or pattern that may be analyzed statistically but may not be predicted precisely.”)

If you haven’t much background in mathematics, don’t worry. Some of the more advanced aspects of analytic investigation do call on what, for most, is arcane, but the basics require little beyond arithmetic.

What is a “formula”? It is an “equation”. And what is an equation? It is simply an assertion that one thing is *equal to* another thing, set down in mathematical form. If you go into a stationery-supply store to buy several pencils, all of the same kind, how much will it cost you? That depends on two things: the price of a pencil, and the number of pencils you will buy. Your cost (here we ignore complications like sales taxes) wil be the price per pencil multiplied by the number of pencils. We can set that down in mathematical form as an equation, where we equate the cost with the price times the quantity:

**Cost = Price x Quantity**

Normally, though, one abbreviates, to keep things looking clean, so that would be written something like this:

**C = P x Q**

There: wasn’t that easy?

In baseball analysis, we write various equations, with the left side being perhaps something like runs scored or games won, while the right-hand side is some set of baseball stats in some combination. We get those equations—those equalities—from a combination of logic and examination of history. And, as always in science (and this *is* science), we then submit our proposed equations to rigorous testing against massive amounts of real-world data.

But at this point, we need to take a little detour to discuss the nature of “baseball equations”. After all, no one is saying that some formula for, say, runs scored is always and ever going to give the very exact number of real-world runs scored. So what use are these equations, and what does “testing them” mean? (This is where the “stochastic” part comes in.) We answer those questions next.

Baseball-related equations are *probabilistic* equations. What does that mouthful mean? Well, there are two basic kinds of equalities in the world: exact and probabilistic. An exact equation asserts something that we believe is always and ever exactly so; the distance a car will travel down a road is its speed (assuming for simplicity that we hold that constant) times the length of time it is driven:

**D = S x T**

Distance driven equals speed multiplied by time. That is exact: do 60 mph for two hours and you will always and ever cover 120 miles, exactly (or as exactly as we care to measure the distance, the speed, and the time). What other sort of “equality” could there be? A *probabilistic* “equality”. If you’re not used to thinking in such terms, that may sound abstruse, but fortunately easily understood everyday examples abound (since the universe we live in is ruled by probability).

Our first tool is nothing more complicated than a coin. We all know that an unbiased coin (meaning one not bent or scraped or very heavily engraved on one side only) comes up heads half the time and tails half the time: that is why we use coin tosses to settle so many things. But what is it that we “know” so surely? No one believes or expects that if you toss a coin over and over that it will come up heads-tails-heads-tails and so on forever; of course not. What we “know” is that as we toss a coin more and more, the percentage of heads (or tails) that we get will *tend* to get closer and closer to 50%.

If we toss a coin ten times and get 6 heads, we think absolutely nothing of it. If we toss the coin 100 times and get 60 heads, we notice it. If we toss it 1,000 times and get 600 heads, we are profoundly suspicious that it isn’t an unbiased, honest coin. And did we toss it 10,000 times and get 6,000 heads, we would consider that iron-clad *proof* that something is wrong with the coin. None of that is abstruse or mystic: it is founded in our ordinary experience of life.

We can write a very simple equation for heads in coin tosses:

**H = T x 0.5**

That says that heads come up as one-half of tosses. That assertion, that *equation*, is the one that we just said everyone knows is true; but we also said that everyone understands that it asserts something that is the *long-term result of many trials*— and that the more trials, the more nearly exact its results are, and correspondingly the fewer trials, the larger we expect the differences between its predictions and real-world results to be.

That is what a “probabilistic” equation is. No one is going to toss a coin 162 times, get 77 heads (or 84 heads) and announce that this heads-from-tosses equation is all a lot of hooey. But if a games-won equation says a given ball club is “a .500 club” and they win 77 or 84 games, some people will say that *that* equation is all just hooey. Obviously, we need some sort of yardstick to decide whether real-world results differ from what a probabilistic equation predicts by so much as to seriously question its validity, or whether they support the claim that it is “correct”.

Fortunately, the wonderful ways our universe works provide easy yardsticks for such testing. Here, we will just present the facts, but, one, you will find that they match ordinary intuition, and two, you can look up the details in many, many places (including a fairly technical article in Wikipedia).

The intuitive relation we cited earlier, where we saw that the more trials (say, coin tosses), the closer we expect results to follow a prediction, happens to be an exact one. That that is so is one of the wonders of our cosmos—that “random” events follow very definite patterns of occurrence—but so it is. The general shape of the relation looks like this:

(Because the curve looks something like a cross-section of a bell, it is often just called “the bell-shaped curve”; its proper name is “Gaussian distribution” or “Normal Distribution”.)

What the graph above shows us is (this is simplified) the likelihood of the results of a given set of trials being just what is “expected”. The horizontal is the value of the result (say, for example, the number of heads in a given set of coin tosses), while the vertical is the number of trails that got that particular result. The peak at the center is just what is “expected” (in our example, 50% heads). Exact correspondence is the most likely result, with differences from that exactness becoming less and less likely as they become bigger and bigger. Wikipedia has a very nice animated display of this, which we reproduce below; in it, we see how the fit to the curve becomes better and better as we take more an more samples (in that image, *n* is the number of samples taken). Give the animation a few seconds to run through its paces from its start at *n* = 4.

As the number of discrete events increases, the function more and more resembles a normal distribution.

To try to ground this in familiar terms, let’s say that we toss a coin 162 times and record how many heads we got. Then we repeat that simple experiment many times. In the graph above, *n* is the number of 162-toss experiments we tried, and the dots sum up our results. The horizontal for a given dot is the number of heads, with 81 in the middle; the vertical is the number of 162-toss trials that had that many heads as a result. As you see, exactly 81 is the most frequent result, and the more we move away from 81—in either direction, higher or lower—the fewer and fewer trials give that result. And the more 162-toss trials we undertake, the more and more our results look like the expected curve.

To say much more would quickly take us into complexity, so we will just go with this takeaway: when we create a probabilistic equation, whether for heads in coin tosses or for runs scored from team stats, there are universally accepted standard tests for the predictions as compared to real-world results from real-world data, and those tests will tell us if our equation seems correct (or not). Those sorts of tests are used every day in everything from political polling to factory assembly-line quality-control sampling to nuclear physics.

An interesting sidelight here is that added data become progressively less helpful in improving confidence. If a rookie bats .300 in his first full year, we are hopeful but not fully convinced; but if he bats .300 or so again the next year, we feel like the team has found a real .300 hitter. If he then bats .300 for his career—well, we already expected that, didn’t we? The increase in at-bats from the 1200 or so of his first two seasons to the perhaps 10 times as many of his full career gave us less new information than the mere doubling of the number from his first year to his second. Mathematically stated, confidence goes up as the square root of the data multiple: that is, it takes four times as much data as we have to double our confidence in what the data are telling us. That’s one big reason why pollsters can determine pretty well, for example, how popular a given TV show is by surveying only a few hundred households.

For instance, if we do the math—which we won’t show here—using typical baseball-team numbers, we find that over a full 162-game season, for runs scored we can expect an average variation from target of a little over 2 percent (about 2.3%) from chance alone (that is using one particular runs-scored equation, the most accurate one currently available). But over a more restricted period of time—say the first six weeks (one-quarter) of a season—the expected average scatter rises significantly, to around 4.6%, owing to the reduced data sample (which is one-quarter of a season’s total, so the error doubles).

The point of this digression is that if you take the results of any competent analysis of baseball statistics—let’s say Owlcroft’s “TOP” formula—and repeatedly compare its predictions against real-world results, you expect to see a bell curve whose exact size and shape depend on definitely known numbers. If that is the case, you have good cause to say that the formula is correct. There are minor differences in accuracy between various different formulae from various different sources, but those differences are very small compared to the degree to which **all** of them, however derived by whom, generally agree with one another and with the expected scatter patterns probability mathematics demands of an accurate formula.

So that you can see that we put our money where our mouth is, we include this demonstration tabulation of **the Owlcroft TOP formula tested** on two-thirds of a century of major-league baseball stats. (You can also take a look at short-term results on the Team-Performance page on this site, but we don’t link it at this specific spot because you should read more before going there.)

We now return to the logic of analyzing baseball.

Individual baseball games are, obviously, won or lost based on a very clear and simple rule: the team that scores more runs than it gives up by the end of the game wins. Less obvious is that there is a definite and clear relationship between the runs a team scores and gives up over a *series* of games and the percentage of games it wins *in that series* (and, again, that of course is a probabilistic relation). Given that fact, if we knew how many runs a team could be expected to score and give up over a season, we could predict with reasonable accuracy how many games they would win in that season.

There are numerous versions of a games-won formula; they often look very different, but when one does an engineering analysis with typical baseball numbers, they essentially resolve into the same thing. Naturally, they thus also each give almost exactly the same results for a given set of games and runs figures. Cook, in *Percentage Baseball*, used simply—

**1/2 x R/OR**

—(where R and OR are Runs and Opponents’ Runs) to get the expected win percentage. Bill James has used his so-called "Pythagorean" formula, not easily reproduced on a web page. We at Owlcroft have yet another. None of them is really right, because “right” here would be a very messy probabilistic equation based on typical scatter of runs scored around its average value for a team (that is **not** a simple bell curve, because it is constrained at one side, the lower limit—you can’t score fewer than zero runs—but there is no upper limit, especially with advent of what we call **the SillyBall**). But they all work quite well enough.

To give you an idea of the problems, run-scoring is not a simple bell curve, because it is bounded by a lower limit of zero runs; thus, the likelihood of runs looks something like the graph below:

To reckon the probability of Team A scoring more runs than Team B when all that differs on their runs graphs is the exact location of the high point (their *average* runs per game) is a far-from-simple mathematical exercise, and none of the extant games-won equations really calculate on this basis: they are useful working approximations that happen to have good accuracy.

(An important aspect of baseball statistics is that in the real world they don’t actually have a very large range of values: for example, once past a few at-bats, no one hits .007 or .731; no team’s seasonal run total is 37 or 23,469; and so on. Because of that narrowness, an equation relating values can be rather drastically wrong overall, but still manage to give tolerably correct answers within that narrow range. To a human, the world looks flat, because the very narrow segment of it we see is curved so slightly we can’t notice it. In mathematical terms, a linear approximation (“linear” meaning a straight line) can work on almost any function if it only has to deal with a very narrow range of values.)

To make this less mystical, consider a team that plays a fairly large number of games against another team or set of teams—in fact, a typical baseball season. If, at the end of that time, the team has scored exactly as many runs as it has given up, it is no great leap of logic to say that on balance they have been neither better nor worse a team than their average opponent. That being so, we would expect that the most reasonable outcome is that they have won no more than they have lost: that they are .500 in those games. All games-won formulae thus must meet the test that at equal numbers of runs scored and runs yielded, they predict a .500 win percentage. (Consider, for example, even Cook’s simplistic formula as given above.)

Moreover, we certainly feel that if the team has outscored their opponents by a little, they should have won a little more than half their games; and if they outscored them a lot, then they should have won at well over a .500 clip (or, of course, vice-versa). The various formulae quantize those expectations, giving specific, reasonably reliable win percentages for specific R and OR run sets.

A point worth noting and remembering is that while thedifferencebetween runs scored and runs allowed is the key factor, the exact criterion is that differencerelative to total run scoring. Prevailing levels of run-scoring are commonly referred to as the “run environment”. If we imagine a league in which the run environment—the average number of runs scored in a game (both teams counted)—is, say, five, then obviously any one run is pretty important, and a seasonal average one-run difference between runs scored and runs allowed will loom large. On the other hand, if the run environment is not five but, say, twelve runs, then any one run in a game means rather less, and a seasonal-average difference of one run isn’t as big a factor in determining likely wins. Virtually all games-won equations relate runs to the runs environment. (Owing to that point, it is not quite exactly true that adding runs on offense or removing runs on defense are equally significant; technically, removing runs—better pitching and fielding—adds very slightly more to winning than adding the same number of runs on offense. We say “technically” because in the range of real-world run values, the difference would only show up for rather huge differences.)

If you want to see how easily one can construct a games-won equation from simple common sense, well, let’s try it. Sheer intuition tells us that the crux is the extent to which a team scores more runs than it allows (or vice-versa), so we start with that difference: **R - OR** (which could be negative). But, as we noted above, the size of that run difference has to be considered in proportion to the run environment; the run environment for a team being just **R + OR**, we now get:

(R - OR) W% = ―――――――― (R + OR)

But that can’t be quite right, because when R and OR are equal, it is just zero. But then we realize that this must instead represent the *difference* between .500 and what the win percentage is, so we just add in that .500:

(R - OR) W% = ―――――――― + .500 (R + OR)

And you know what? That actually works pretty well. It turns out that by “tuning” it to real-world values, we can improve its accuracy a bit by multiplying the runs-ratio part by an arbitrary constant a little under 1, but the improvement is small: our original was already pretty good.

Now a careful reader who happens to know a little algebra will probably notice that this equation cannot be “correct”, because at an R/OR (or vice-versa) ratio of 3 or more it predicts win percentages over 100% (or under 0%, as may be). And that is so. But we already said that *none* of the extant equations are “true”, only engineering approximations; within the narrowly limited set of run values one finds in the real world, this little toy works as well as or better than any of the others, and we can see just where it came from in logic.

Indeed, for many years we used that very equation (with a carefully reckoned “fudge factor” constant) and got good results. But, in our recent upgrade, we switched to a more commonly used “Pythagorean”-type formulation that gives better results—*triflingly* better, but we try for as much accuracy as we can get, though rounding calculated wins to whole numbers almost wholly erases the difference in results. For the record, that formulation (where “**^**” signifies “raised to the power of”) is this:

(R^1.8484) W% = ―――――――――――――――――――――――― (R^1.8484) + (OR^1.8484)

Its average error over 1,662 team-seasons (every team for 1955 through 2019, inclusive) is less than 2 wins per team-season: 1.8532096462704 to be exact (which, of course, rounds to 2 as a whole number). Here is a graphic display of those results (with all projections rounded to whole numbers):

If we can—and we indeed can—project probable games won from runs scored and runs yielded over any arbitrary set of games (with, of course, increasing accuracy as the number of games in the series rises), we would next like to be able to project runs scored and yielded for a team based on who is playing and pitching for it. If we could do that as well—and here, too, we can—we could then project with some accuracy a team’s ultimate win percentage just from the identities of its player personnel.

The essence of scoring runs in baseball is remarkably straightforward: put runners on base and then drive them in. The background is the ticking clock of baseball—outs. Of all the many and diverse numbers in baseball analysis, none is nearly so important as this one: * three*, the three outs that define an inning.

(That observation, published by us in 1981 inThe Sinister First Baseman and Other Observationslater led George F. Will, writing inThe New York Times, to refer to us, by name, as “the most important baseball thinker you have never heard of.”)

But to continue: another thing that probability mathematics tells us is that the chances of two things *both* happening is the chance for one multiplied by the independent chance for the other.

To see that, imagine tossing a quarter and rolling a standard six-sided die (at the same time or one after another, it doesn’t matter). What are the chances of getting a tail from the quarter and the number 5 on the die? The chance of getting tails on the quarter are one in two, or 0.5. No matter what the die does, in half the cases the quarter is heads, so we can only get the right die roll in half the trials. The chances of getting a 5 (or any particular one of the six faces) are 1 in 6, or about 0.167. But since that only “wins” in half the cases, owing to the quarter, the net odds are 0.5 times 0.167, or about 0.083.

The chance of a man getting on base is very simply expressed by a now-familiar (if late-arriving) stat: the **on-base percentage**. To get the chances of a man at the plate becoming a run scored, we need to take his on-base percentage and multiply it by some factor representing the chances that a teammate will knock him in. (We do need, naturally, to make some adjustment to the raw on-base percentage to allow for the facts that the man may get on by an error, and also that he may be put out on the basepaths even after having reached safely).

As an aside, we need to remember at all times—which many discussions and analyses we have seen do not—that ** the batter at the plate is also a base runner**. That is, there is

The mechanics of what such an “RBI factor” might comprise, and of how it is derived, are somewhat complicated. Evidently base hits are going to be very important, and extra-base hits especially so; but walks have some value, and even minor factors like wild pitches and balks are not utterly negligible. In fact, the details of both the philosophy and practice of calculating an RBI factor of some sort are largely (but by no means wholly) what distinguish one school of analysis from another. Owlcroft has its own methods, which we will not detail here for a variety of reasons, most notably brevity.

In many workers’ formulations, the occurrence rate of Total Bases (the sum of all hits weighted by bases per hit—that is, for example, triples are 3 and singles are 1) is the only determinant in their RBI factor, whatever they call it (if they call it by a name at all). That can actually give a pretty fair result, and it has the virtue of simplicity. The first runs-scored formula Bill James widely published was indeed that simple:

**(Hits + Walks) x (Total Bases) / (At-Bats + Walks)**

Since **Hits + Walks** is, roughly anyway, the available base-runner total, and **At-Bats + Walks** is—also roughly—the plate-appearances total, manifestly James’ “RBI Factor” in this formulation was indeed just the Total Bases rate (TB/PA, more or less). Note that James did not use the on-base percentage, or any rough equivalent of it: he used what amounts to the actual number of base runners. That’s OK for a quick, simple formulation which will serve to demonstrate how well analytic methods work, but it limits the utility of the formulation to evaluating what *has* happened; you cannot use it to predict what likely *will* happen because to know how many men *will* reach base, you need to state your formula in terms of an on-base *rate*. And that brings us to another important point.

The on-base percentage and an RBI factor, when multiplied, give the **chance** that a given batter will become a run scored; but the actual **number** of runs scored also depends on how many men come to the plate so as to *have* that chance. That number, actual total team plate appearances, varies significantly from team to team and year to year; but it does not do so without cause. Remember outs as the ticking clock of an inning: the less likely a team is to make an out at the plate, the more men they will get *to* the plate over the long haul. That can be stated quite precisely in a mathematical formulation, but its essence for the purposes of understanding is this: **a team’s on-base percentage has a form of “compound-interest” effect on run scoring.** First, it directly increases the chance that any one batter will ultimately become a run scored; but second, it increases the number of men who will get to *have* that chance. It is for these reasons that the single most important baseball statistic viewed in isolation is, far and away, the on-base percentage; actual run scoring tracks on-base percentage more closely than any other single statistic (as we now understand that it should).

Let’s construct a simple runs-scored equation, just to show the logic of the process; the one we now describe is one we published almost four decades ago, at which time we described it as “a very-much simplified version” of a full run-scoring equation. It looked like this:

(H + BB) x TB R = ―――――――――――――――――――― x G x 27 x 0.93868 (AB - H) x (AB + BB)

Let’s see what that breaks down into. As we noted above, **H + BB** is, roughly, men getting on base; also as noted above, **AB + BB** is, roughly, total plate appearances. **TB** is just Total Bases, which we are using as our “RBI Factor”. Beyond those, **AB - H** is (again, as always in this crude version, “roughly”) outs made. The **G x 27** is simply all seasonal outs (which we approximate as 27 per game, not exactly true but quite close) times an empirical “fudge factor” of about 0.93868 (more exactly, 0.93868463330814).

Now let’s look at that equation in another way:

**R = { (OB/PA) x (TB/PA) } x PA**

That is the simple idea stated above that the *chances* of a batter becoming a run are the on-base percentage (**OB/PA)**) times an “RBI Factor” (here, simplified to just (**TB/PA)**), and that runs are those *chances* of a man becoming a run times the number of men who *get* that chance (**PA**).

All we have done is to note that PA, plate appearances, is simply all available outs (27 x 162, or 4,374) divided by *the chances of making an out*, which is simply the outs *rate*, Outs per PA, which is just **(AB - H)/PA**).

As we said before: there—wasn’t that easy?

Note that that equation uses only *four* stats: AB, BB, H, and TB. The empirical part (that 0.93868 multiplier) tries to roll up everything else, from men thrown out on the bases to sac flies and a ton more. Yet, for all the simplifying and ignoring that goes into such an elementary version, its accuracy over 3,324 team-seasons (every team for 1955 through 2019, inclusive, applied to both offense and defense stats) is 96.54% (that is, the average error is 3.46%, which compares decently with the full-force equation’s 97.71% accuracy, an average error of 2.290%). It was and is remarkable how good one can get with even the simplest of formulae.

(What the “full” version does is introduce the complications: extra-base hits are weighted more accurately than the simplistic “1-2-3-4” weighting, things from walks to sac flies are introduced into the “RBI Factor”, each with an appropriate weighting, and suchlike. The mechanical details are messy, but the principles are simple and obvious.)

And one more time: if you have any doubts that the Owlcroft run-scoring equation works, and works very well, look over **the actual results** again.

Now consider this: what we can calculate for a team from its statistics, we can also calculate for *any one batter* from his personal statistics. If we then set the number of available outs to what it is for a full team for a full season, we get a number that sums up that man’s ability to contribute to his team’s scoring of runs in one number; we can think of it as *the runs that would be scored in a season by a team made up entirely of exact clones of that man.*

Owlcroft calculates just such a measure, which we call the *Total Offensive Productivity*, or just **TOP**. It is shown for all batters listed anywhere in these pages; in the by-team batting lists, the batters are arranged in order of descending TOP.

Moreover, what one can calculate for a batter, one can correspondingly calculate for a pitcher, using the numbers that he gives up to batters. You will find on this site just such calculations, which yield a novel and very, very important measure that we call the
**TPP**, for *Total Pitching Productivity*, because the term pairs nicely with the TOP). We also generate a measure we call the “Quality of Pitching” stat, so those accustomed to using ERA as a measure can have something comparable to review. There is **a separate page on this site** that discusses those measures further, but you would be best off to finish this page before jumping there.

Two other and somewhat related points need mention. (Actually, they need extensive discussion, and we hope in future, as we gradually expand these notes, to give them that discussion.) One is the predictability of individual men’s performances. As we said earlier, there is a sort of law of diminishing returns for the meaning of increasing data; by the time we have roughly the equivalent of two or at most three seasons’ full-time play for a batter, we have enough data to have defined his norms of performance pretty well. Pitchers, for complex reasons, take more time to evaluate, although using the TPP instead of the ERA gives results in time periods comparable to those needed for batters. Moreover, by the time a batter reaches double-A ball, he has become pretty much what he will be; if we have two or three seasons’ worth of data **above class A ball**, we have the man defined. That is not true for pitchers, because batting is responsive, while pitching is initiative: a pitcher can, even late in his career, add a new pitch or other change to his delivery, whereas a batter can by and large only work with the reflexes nature has given him.

It is precisely that predictability that makes it possible to “engineer” a baseball team in a manner quite comparable to the process of engineering an automobile engine. By knowing the data for the components and the equations for how those components interact, we can design an engine or a team to meet a specified set of performance criteria. It is crucially important to understand what we are saying here: we are ** not** saying that we can predict accurately how every man will do in a given season from how he has done in the past; that is, as common sense suggests, impossible. But, just as we certainly cannot predict how a pair of dice will come up in any given throw or small number of throws—which is why people gamble—we can equally certainly predict with great accuracy how much money a craps table will likely take in on one shift because we know the

The second point, related to what we just said, is that minor-league statistics—long thought by most baseball professionals and fans to be nearly meaningless—* can* be translated so that we see what the man would have achieved playing at that same level of ability in a major-league ballpark against major-league competition. (That realization, and the mechanics to implement it, are one of Bill James’ most valuable contributions—probably his

Finally, we repeat that all statistics, to be useful, must be comparable. There is a separate page on this site that discusses **“normalization”** processes for stats and why they have become a statistical nightmare (we no longer apply them, preferring honestly raw results to results “adjusted” by such dubiously derived factors).

A wholly other way of reckoning runs scored from a team stat line is something called “Linear Weights” (LW). It works on the implicit assumption that results from any one datum—say singles—affects run-scoring on a straight-line relation (hence “linear”), so that runs can be calculated by using a set of weighting constants (hence “weights”), looking something like this:

**R = (k1 x 1B) + (k2 x 2B) + …**

The various weighting constants can be gotten in many ways, mostly empirical. We can also, in principle, get them from a run-scoring equation of the probabilistic kind described farther above. We just lug in all-MLB-average team stats, then see what effect on the runs total comes up if we add one more, say, single (something often called the “plus-one” value). That, in principle, gives the weighting factor for whatever stat we plus-one’ed (such as singles).

How to do that isn’t as obvious as it might seem at first blush. Do we add one more PA and assign it to a single? Or do we change one existing out to a single? And either way, what about the change in the PA total that arises from improving the OB%? Points to consider…

One issue we see with LW, which we do not use or elsewhere refer to on this site, is that the value of, say, a single is not a fixed thing: it will mean more on teams with more power and higher OBPs, because a man who gets on has a better chance of scoring on such teams, and conversely less for weaker teams for analogous reasons. Some people love LW; we don’t. We prefer measures that have an underlying logic and are not 100% empirical.

As any modern baseball fan knows, there is also an entire alphabet soup of other metrics out there, from fWAR to UZR to xFIP and so on and so forth. These have value, some more than others, but there are two points about them as a class that especially bother us.

One is that many of them, notably the “WAR” stats (“Wins Above Replacement” if you didn’t know) is that they are not *rate* measures but *cumulative* measures. If I am thinking of buying a racing car, I mainly want to know how fast it goes; I am not especially interested in how many miles it drove in the last week or month or racing season. While if I were to examine its history I could probably derive, in time, its average speed in driving those miles, that is a roundabout way of getting to what should be a single, simple stat. I don’t usually care how many WAR a player has in a given season, because I did not control (and may not know without some looking-up) how often the player played, and why he wasn’t playing when he wasn’t (hurt? platooned? manager’s dog house? something else?). And, of course, another issue is the entirely artificial and arbitrary “replacement-value” standard; back at the race-car analogy, do I care about miles driven in a racing season above those of some imaginary “replacement car”? Really?

A second issue, possibly even more pernicious, is that the modern alphabet soup is mostly or wholly *relative* measures. Things like games won or runs scored are results that correspond to real-world facts: we can (a) test any equations for them against real-world data and prove (or falsify) our hypotheses, and (b) easily understand and use those results. But with *relative* measures, we have slip-slided all the way back to pre-analysis days, when taverns were filled with baseball fans arguing over whether this guy’s batting average made him more valuable than that guy’s RBIs, and no real way (save having the louder voice) to settle the argument.

It was a very long and wholly uphill battle to establish credibility for analysis in baseball, and the war is far from over. To revert from real, absolute numbers to arcane and wholly relative measures just re-establishes in old-timers minds the idea that all this numbers stuff is just a fancified set of tavern arguments. Sure, an analyst can add up a team’s players’ and pitchers’ WARs and get a team number, but it’s a number founded on an entirely imaginary and arbitrary team of “replacement-value” players. “Find and point to that team, and show me in print that team’s wins,” might any baseball executive say. And what is the reply to be?

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