THE BOOK--Playing The Percentages In Baseball

Another way to look at this is the spread in talent for a fixed number of players.  Now remember, what I did earlier was based on the number of teams, so in effect what I was giving you was the spread in talent per team.

Now, I’m going to take a fixed number of players, 200, and tell you how much the spread in talent has changed in that group.

From the early 1900s through until WWII, the spread in true OBP among the top 200 players in PA, was in the .040-.045 range.  From that point onwards, until 1958, it’s been pretty flat at .035-.040.  Since then, until around 1980 it’s been in the .030-.035 range.

Afte that point, the spread in talent among the 200 regular players has been increasing a bit every year, with a sustained peak from 1993-2002.

It is important to note that I did not control for park factors, and introducing Coors into a study certainly requires one to do so.  Coincidentally, since the PED years happens to coincide with this time period as well, it certainly bears further investigation.

That is, while we expected a tightening in talent levels, as the years go by, since we are drawing from more players, and more people want to be ballplayers, and we are always looking at 200 players, we instead find a reversal in the last 10-15 years.  (A reversal that seems to have unreversed itself suddenly in the last couple of years).  Whether that has to do with PED, Coors, all the new parks, or a different selection process for players, more work needs to be done…

It would also be interesting to correlate the impact of league expansion with changing spread in talent ...

Do you have a standard deviation for the “spread in talent” per year (the standard deviation by chance of the estimated standard deviation of skill)?  We certainly want to know the uncertainty of your measure.  For example, how signficant is the difference between the .027 and the .033?  Inquiring minds need to know that.

That’s next for me to do.  I gotta re-re-re-re-read our Appendix again… easily the most re-read piece in The Book for me.

One thing though is the Bonds-Williams-Ruth effect.  These guys are such outliers, even among the outliers that are MLB players, that it significantly skews the distribution.  If you look at their career z-scores, I think they are at 6 or 7 or so.  I’m thinking that at the least, I should artifically max out the z-scores of the players, since we are looking at the spread in talent, and having such a point that is really out there, really stretches the distribution to make it (try to) fit that point.  I am including the IBB, so that may be another issue to consider.

I am not sure that you are really measuring a spread in talent but rather a change in philosophy of fielding.  The standard deviation of regular players in any given year would decrease if teams were making a decision to sacrifice defensive ability at the traditionaly defensive positions (SS, 2B, and CF) for offensive ability.

Tango

This is a very interesting post and certainly got me thinking about standard deviation, luck and skill.

Just one quick question. By pre-selecting the best players (ie, point 2 in your initial recipe), you could be introducing selection bias into your estimates as you’d expect the better players to garner more player time. So the question is does the sample that you have selected approximate to a normal distribution? My guess, without replicating your analysis is that it probably does but I was wondering if you had checked?

John

Whatever bias I would introduce, would apply to all the years, right?  I’ll check anyway.

Peter has a valid point that there are other reasons that the spread in OBP might change.  I was thinking of it only in terms of other hitting elements, but it could be other player components as well.

I took all the teams since 1962, with a minimum of 158 GP (n=1022), and calculated the standard deviation of their actual (observed) winning percentage.  That’s .072.

I then figured the random variation, expected, given 161.8 trials (the average number of games per team in my sample).  That’s .039.

Since…
variance(observed) = variance(true) + variance(random)

...we can calculate the standard deviation as .060.

That is, the true spread in talent, at the team level, over the last 40 years or so, is .060 wins = one standard deviation.  (If all players were randomly placed on teams, we would have expected the variance of the true talent to be .060.)

***

The observed standard deviation for runs scored per game is .58, and runs allowed is .59, which is around .015 runs per PA, or .013 wOBA. 

That is, 50% of baseball is offense, and 50% is defense.

***

With each team having around 6000 PA, our random standard deviation is .0065, making the true wOBA .011.

That is, the variance of the players’ hitting, running, the parks, gives us true wOBA of .011.  That’s true on the offense and defensive side.

.011 ^ 2 = offense ^ 2 + parks ^ 2

Assuming that parks = .004 (that is, the standard deviation of the parks is .004), then players = .010

What does this mean?  That the true spread in talent, offense, is one standard deviation = .010 of wOBA.

For the defense, you have two forces, the pitchers and fielders.  For the moment, I’m going to assume that the spread in pitching talent is a bit larger than the spread in fielding talent.  In that case, we get:
.011 ^ 2 = pitching ^ 2 + fielding ^ 2 + parks ^ 2

That gives us pitching = .008, fielding = .006.

Anyway, going back to the hitters.  If the true spread at the team level is .010, then what’s the spread at the player level?  Assuming nine full-time hitters:
.010 ^ 2 = (h1/9 ^ 2 + h2/9 ^ 2 + ... + h9/9 ^ 2)

h1=h2=...=h9=.030

That is, the spread in individual hitting talent is .030 wOBA.  Remember what my original blog entry said?

The tightest distribution in OBP talent was 1978, where one standard deviation = .025 OBP.  That is, we expect that in 1978, 95% of all the regular players had a true talent rate within .050 points of the mean.  The widest distribution, ignoring the early years, was 1923, with 1 SD = .042.

Well, the average for 1962-2005 was .031 of OBP, which is .033 of wOBA.  Since I didn’t account for parks, the .033 of wOBA with no park adjustment is equivalent to .030 of wOBA with park adjustment.

So, there you go.  Two completely different approaches, to tell you that the spread in offensive talent is .030 of wOBA (or .028 of OBP).

Repeating the calculations for the pitchers (assuming 9 equal pitchers of 162 IP each), and we get a spread in pitching talent of .024 wOBA.  Fielding talent is .015 wOBA.

However, if we separate the pitcher as fielder from the other eight nonpitchers as fielders, the pitcher-fielding talent is probably .010, and the 8 nonpitcher-fielding talent is probably .019.

So, the final spread in nonpitcher talent is:
sqrt(.030^2+.019^2) = .036

And for pitchers, it’s:
sqrt(.024^2+.010^2) = .026

That is, 58% of baseball is nonpitchers, and 42% is pitchers.

(Given all my assumptions, regarding parks and the initial pitching/fielding split.)

Very interesting, Tango.  A few questions/observations:

What is the basis of the pitching fielding split of .008/.006?  Just a guestimate, or based on research? 

How can you use team level offense variance of .01 to calculate individual player variance of .03, when the quality of player1 is not independent from quality of player2, etc.?  If teams were assembled by letting team1 pick the 25 best players, then team2...with team30 getting the 25 worst players, you would have the same underlying player talent spread but a much larger team spread.  Presumably, teams that are richer, smarter, etc. will have more good players than other teams. And this may not affect hitting and pitching in the same way:  one could imagine that hitters’ greater predictability could mean that offensive ability is more “efficiently” concentrated in certain teams. 

Similarly, when you say “So, the final spread in nonpitcher talent is: sqrt(.030^2+.019^2) = .036,” doesn’t this assume that hitting talent and fielding talent are independent?  In fact, I would think they are negatively correlated (good fielders = weak hitters), so the spread of total nonpitcher talent should narrow.

Guy,

To your first point, since the results of assuming that hitting talent is independent produces a spread consistent with another method that assumes independence, we have a pretty good check here that hitting talent does not cluster.

As to your second point, you are right, that I should have made that clear, that I treated the two as being independent.  If they were totally dependent, then the spread would be much narrower (.023).  My guess, as yours, that there is some level of interdependence.  However, I seem to remember doing this work, and finding that it wasn’t that strong.

For the moment, I just left it as a guesstimate, with limited research.

Pitchers are involved in 25% non-BIP, and 75% BIP.  But, the true variance in the 25% is much larger than in the 75%.  For fielders, they get 0% in the non-BIP, and a portion of the 75%.  Luck gets a little portion in the 25% and a large portion of the 75%.  The 8/6 split just seems reasonable, though you could easily make the case for a 9/5 split.

I thought the reason Win Shares undervalued starting pitchers is because it places identical value on runs prevented (defensive marginal runs) and extra runs scored (offensive marginal runs). A run prevented is considerably more valuable than an extra run scored.

No, Win Shares undervalues pitching because it doesn’t understand the following three truths:

1. When given two variables, offense and defense, offense is 50% of the game and defense is 50%.

2. When given three variables, offense, pitching, and fielding, offense is 42%, pitching is 33%, and fielding is 25%.

3. When given two variables, nonpitchers and pitchers, the nonpitchers are worth 58% and the pitchers are worth 42%.

I have explained these two seemingly contradictory statements elsewhere on my blog (do a search).  Nonetheless, the summary is that you cannot try to split “absolute” numbers.  It simply won’t work.  The basis must be on the spread in talent (standard deviations or SD), and everthing does add up (the variances, which is SD squared).  That’s the whole problem.  The SD squared add up, but the allocation is based on SD.

Please, do a search, so I don’t repeat myself.  If you can’t find it, I’ll look it up.

Here’s one relevant post:
http://www.insidethebook.com/ee/index.php/site/comments/spread_in_talent/#8

Obviously, the statements aren’t contradictory, and I think they actually support my point, which is that in MLB saving runs is more valuable than scoring runs. Or, as you say, pitching and fielding add up to 58 percent. But in Win Shares defensive and offensive marginal runs have almost the same value. If that system increased the value of defensive marginal runs to account for 58 percent of wins, pitchers would have many more Win Shares—20 to 25 percent more.

I also think Win Shares sets the zero-level too low, but I’m not sure how that affects the distribution of shares.

I’m a little confused on the breakdowns and I’m not sure what I should be searching for in the blog to clear this up.  If when given 2 variables, offense = 50% and defense = 50%, then why when given 3 variables, offense = 42%, pitching = 33% and fielding = 25%.  Why doesn’t pitching and defense sum to 50%?  What am I missing here?

I don’t quite understand how fielding + pitching = 58% and defense = 50%, but I’m not going to argue with you on it.  I think I’ve been through the explanations before and it makes sense for a little while, then I forget about it.

Does it depend on position players not being selected solely for offense or defense, but a combination?

In other words, if we had a lineup of 9 DH’s, like football, and 8 separate defensive players + the pitcher, would fielding + pitching = 50% ?

Rally/33: perfect example.  If you had a football-like setup in baseball, then the spread in runs scored will almost certainly be lower than the spread in runs allowed.  It may not work out to 42/58, but it’s probably be close to that.

And yes, in that case, pitching+fielding would = 58, because all three components (pitching, fielding, hitting) would be independent.  And off=42 and def=58.

In real MLB, it’s because that nonpitchers have a hitting spread and a fielding spread that you get the issue.  You can add the variances, but the spread in talent is based on the standard deviation.

Tango/32—You’re right, but the glory of investing in pitching is that you’re also investing in lowering the run scoring environment at the same time.  You can’t allow fewer/more runs without changing the RPG.  Since pitchers determine their own environment, I don’t have a problem with crediting them for more wins per run saved the better they are.

sky/36: Sure, and I do that, since I put the pitcher through PythagenPat to get his win%.

But, at the team level, you can’t imply that off/def is anything other than being very close to 50/50.

tango 32: “wes/30 : read posts 19,20, then let’s talk.”

****

Ok, been there, done that.  Now I’m ready to talk again.

I see how non-pitchers contribute more than pitchers, since non-pitchers also field.  I get where a run saved becomes worth more than run scored when you add enough to change your run environment.  I also understood Rally’s football example to a certain extent.  An even more extreme verion of that might be home-run derby with a pitching machine???

But I still don’t see how we can observe that the SD of runs scored is the same as runs allowed, then say that the offense/defense split is 50/50 and then finally conclude that pitching+fielding> defense.  Sorry if I’m being thick here, but I really don’t get it.

Elsewhere, I wrote the following:

1. When given two variables, offense and defense, offense is 50% of the game and defense is 50%.

2. When given three variables, offense, pitching, and fielding, offense is 42%, pitching is 33%, and fielding is 25%.

3. When given two variables, nonpitchers and pitchers, the nonpitchers are worth 58% and the pitchers are worth 42%.

I have explained these two seemingly contradictory statements elsewhere on my blog (do a search).  Nonetheless, the summary is that you cannot try to split “absolute” numbers.  It simply won’t work.  The basis must be on the spread in talent (standard deviations or SD), and everthing does add up (the variances, which is SD squared).  That’s the whole problem.  The SD squared add up, but the allocation is based on SD.

I will try to explain more here.  In post 8, I said this:

The observed standard deviation for runs scored per game is .58, and runs allowed is .59.

That is, 50% of baseball is offense, and 50% is defense.

This is what we talk about when we try to split up “percentages”.  We are in fact talking about the spread in true talent.  For example, imagine a game where you are playing against pitching machines, and you either hit a HR or are out.  Each pitching machine has its own serial number, but are all manufactured by the same company.  You’ve got 30 pitching machines, and 30 MLB hitters.  Each plays 162 games of 40 at bats.

The observed standard deviation of the runs allowed by the pitching machines will be EXACTLY equal to what you’d expect from random.  While some pitching machine will be the “leader”, it will be so purely by luck. 

The observed standard deviation of the runs scored by the 30 players will be a combination of luck and their true talent.

You can infer the true talent spread by this equation:
variance(observed) = variance(true) + variance(luck)

For the pitching machines, variance(observed) = variance(luck), leaving variance(true) = 0.

And that makes this form of baseball 100% hitting.  We on the same page so far?

Thanks for extending the HR-Derby w/ pitching machines example.  You’ve painted the picture exactly the way I imagined it would be.  So yes, we are on the same page, so far.

You can now extend the same example, but now you have robots on the field of play as well (or cutboard cutouts).  If a ball hits the cutout (which will be 10 feet high, 10 feet wide, and 10 feet long in the IF, and 30/30/30 in the OF), the batter is out.  It lands anywhere, and the batter gets a one-base hit.  He’s still doing this against a pitching machine.

Once again, we are still at 100% offense, 0% defense.

Cool so far.  I see the cutouts now.  Carew, Boggs and Gwynn are seemingly hitting line drives just barely to the left or right of them.

Now, let’s replace the pitching machines with real pitchers.  But, these pitchers are only allowed to throw fastballs, and are only allowed to throw them at between 75mph and 80mph.

So, we are removing from a pitcher his skillset in mixing up pitches, and even having more than 1 pitch, and resorting his skill to purely be one of location.

We expect that the true talent standard deviation of our 30 pitchers to be a bit greater than zero, but still nowhere close to the true talent standard deviation of our 30 hitters.

Let’s use our measure as outs per PA.  The standard deviation of outs per PA will be fairly tight for pitchers, say 1 SD = .001, while it will be wider for our hitters, say 1 SD = .010.  That is, the spread in hitting true talent is 10 times wider than that of pitching true talent.

The variance is SD squared, meaning the variance for pitchers will be .000001 and the variance for hitters will be .0001 (i.e., the hitting variance is 100 times bigger than the pitching variance).

What we care about here is the SD not the variance.  And, in our case, the spread in talent is 10 times bigger in hitting, which makes this game of baseball 90.9% hitting and 9.1% pitching.

Tango, why do we care about standard deviations and not variance? I’ve always been on-board with this idea, but I actually can’t remember why we want to use standard deviation instead of variance.

I have no opinion on David’s question of SD vs variance, but either way, I’m still with you.  We haven’t gotten to the point where this breaks down for me yet.

What is standard deviation?  It is, essentially, the average difference of points from its mean.

For example, if you take a population of numbers, from 0 to 100, each unique, the mean is obviously 50.

If you take each one, then 0 is 50 from the mean, 1 is 49 from the mean, etc.  The average of the absolute differences of these 100 points is 25.

The standard deviation is 29, which is a number very close to the average of the absolute differences.

(In this case, we had a uniform distribution of data points.  If we had a normal distribution of data points, then our SD would have matched our average absolute differences.)

But, what’s the variance of our data?  It’s 850 (the SD squared).  What does that tell us?  All it is is the average of the squared differences.

More to the point, if the unit of your data point is say “salary” or “on base percentage”, then the standard deviation or the average of the absolute differences is in units of salary or OBP.  The unit of variance is salarySquared or OBPsquared.

And that really doesn’t represent the unit you are trying to measure.

I had assumed that true var(all) = var(hitting) + var(baserunning) + var(pitching) + var(fielding), and so we use variance figure the percentages, not SD.

You use the variance, because that is the equation.  The equation IS:
all^2 = hit^2 + run^2 + pit^2 + fld^2

Just because the equation allows you to add the variances does NOT imply that when you split the credit that you must stick to the square of the standard deviation.  After all, the units of those things would be runsSquared.  And that’s not what interests us.

I’m not sure I follow or agree, Tom. I think you need a more rigorous mathematical proof here.

Why not do SD^3 or SD^10?  It all gets back to the units.  Standard deviation is the spread in the exact unit of what you are looking at. In our case, that’s runs.  Why would you care about runs squared or runs to the tenth power?

Right, but mathematically, it does not necessarily make sense to say that a facet’s share of credit should be it’s SD divided by the sum of SD for all facets. Intuitively, it makes more sense that it would be variance (since all variances must add up to the total variance, though frankly the results we get from adding up SDs seem more right). The right answer shouldn’t be too difficult to derive mathematically.

So I was hoping that this thread was going to go a little further.  Is there a plan to follow-up.  Has there been any re-evaluation of the process in light of David’s comments?

There’s really no point for me to go on, until everyone else gets a satisfactory answer to the SD v Variance question.  To me it’s obvious that you need to keep everything in the unit you are measuring, and that means SD.  The property of variance (SD squared) is such that the variances add up.  But, that by itself doesn’t mean that *all* your calculations must be based on the variance.

The spread is measured in units, not units squared, and I contend that your share must be based on the units, not units squared.

But doesn’t it make more sense to say “pitching accounts for 40% of the variance, hitting accounts for 30% of the variance, etc.” If what we’re interested in is percentages…and the individual variances add up to 100%...and the individual SDs do not add up to 100%...then variance sounds like the right thing to use.

It does sound like it, except it’s not.

Say that the average hitter is 1 SD = 10 runs, the average pitcher is 1 SD = 8 runs and the average fielder is 1 SD = 6 runs (made up numbers).

The variance would say:
100 + 64 + 36 = 200, so that hitting is 50%, pitching 32% and fielding is 18%.

First you have to explain to me how the pitching SD can be 80% as wide as the hitting, yet account for 64% of the value.  (You squared it, but that’s meaningless to me.)

Next, combine the SD for nonpitchers (hitting + fielding) as 1 SD = 11.7 runs and pitching remains at 1 SD = 8 runs.

Once again, I look at that, and I think that pitching should be 68% of nonpitching.  But, a squaring of the results makes it 47%.

But, the units we care about is runs, not runsSquared.  We pay by runs, not runs squared.  Every relationship we have is based on runs.

Just because a PROPERTY of standard deviation is that by squaring its values (AND its units) lets you add things up, doesn’t mean that you want to base anything beyond that on the square of the units.

After all, what does “one variance = 100 runs squared” MEAN?  I mean, to me, it means nothing at all.

I agree with Tango that SDs matter here, not variance.  I think the best way to think about this is value above replacement, denominated in runs/wins (the units we care about, as Tango says).  If we say that replacement level is -1 SD, then the pitcher share of value will be SD(p)/(SD(p) + SD(non-p)).  I assume that gives us something close to Tango’s 42%.

Now, the shares for hitting and fielding MUST add to 58%.  If non-pitchers contribute 58% of the value above replacement, fielding and hitting cannot add to more than (or less than) 58%—this is a mathematical truism.  So a 43/32/25 split based on three SDs cannot be correct. 

How do we square this circle?  By recognizing that non-pitchers will sometimes have negative fielding value or negative hitting value, because they come as packages.  A great hitter may provide sub-replacement hitting, or vice-versa.  In other words, you can’t look at the SD for fielding and use that in isolation to calculate the total amount of fielding “value.” Doing that makes the assumption that there is something called replacement-level fielding which is -1 SD.  But in reality, you can basically get league-average fielding for free!  The key point is that fielding and hitting are not independent.  A -1 SD fielder will invariably be a decent hitter, and a -1SD hitter a decent fielder.  We have to take their combined contribution to measure the value of the player, and it’s THAT SD which gives us the amount of value above replacement contributed by non-pitchers. 

If baseball used football rules, THEN using the 3 separate SDs would be correct.  But then, the hitting and fielding SDs would both be noticeably smaller (a lot of minor lg 1Bman would have jobs as MLB hitters; a lot of minor lg SSs would have jobs as MLB 3Bmen and 1Bmen; minor lg CFs would play RF and LF).  The total value from hitters and fielders might be more than 58%, but certainly less than what you get using today’s 3 SDs. 

Now, dividing the 58% into hitting and fielding is a challenge, and I’m not sure there’s a “correct” answer.  You could use the ratio of the two SDs, and determine that it’s 36% hitting and 22% fielding.  Personally, I think it’s significant that the average fielder is only slightly better than a replacement player in the field, but much worse at the plate.  So I would use the run gap between average and replacement (maybe 18 runs for hitting, 3 runs for fielding), which gives you a division of something like 49% hitting and 9% fielding. That sounds like a low % for fielding, but much of the variation we see in fielding only exists because players are selected mainly for their hitting skill.  Average fielding doesn’t have a lot of value just because teams use worse fielders; the fact is it’s freely available.  In fact, we should say that below-average fielding has negative value—it’s a price that teams agree to pay in exchange for the offense these players provide.  By allowing for negative fielding value, you can have a system that allows for some fielders to be worth far more than others, while still having a total fielding value that’s quite small, as it should be.

I guess I’m still bothered by the fact that we’ve demonstrated that the SD in runs (and/or variance)by the offense and defense are essentially the same, which would suggest that the shares for offense and defense should be equal.  Since hitters don’t prevent runs and pithcers and fielders don’t score runs (when they on defense), it seems to me that picthcing and fielding have to add up to a max of 50%.

Tango/38:

It’s not that I necessarily disagree as much as I’m just trying to get my doubts out there and figure them out. What I’m wondering is that since variance is always expressed as the square of a unit, it’s never in the same unit as what you’re measuring. So then when is variance meaningful, if the unit it’s expressed in is, by definition, meaningless?

I find variance meaningless except when adding them. 

It’s also required to measure correlation (r-squared, coefficient of determination), but even then, I stick with r.

Guy #39, good post!

Thanks, David. 

I notice now that the 3rd sentence in last graf is garbled.  It should read:  “Personally, I think it’s significant that the average player at any given position is only a slightly better fielder than the average replacement player, but is a much better hitter than a replacement player.”

I have a rather fundamental baseline question.  When we’re split up the percentages here, (e.g. offense 42% : pitching 33% : fielding 25%), what exactly are these percentages of?  In post 13, Tango is drawing a compariosn to Win Shares, but I don’t think that’s what we’re talking about here.

Last night I was reading the chapter “44 Percent of Baseball” in the Hidden Game.  Palmer actually gives about a half dozen possibilities of what this could mean and actually refers to it as a confusing and silly question.

I have a guess as to what we’re talking about in this case, but I’m interseted to see how others are defining this.

The percentages refer to the standard deviation, the actual spread, the absolute differences.

If you stick to runs (numbers for illustration only), and you have:
hitting: 1sd = 10 runs per 162G
pitching: 1sd = 8 runs per 162IP
fielding: 1sd = 6 runs per 162G

Then, the conversion to percentages is simply 10/24, 8/24, and 6/24.  That is the thing it does.

I guess I see what you are doing.  I’m just not sure I understand why it makes sense to do this?  What does the 24 represent?  The standard deviation of something??  I’m not convinced that our SD’s, while measured on the same scale R/162G, actually mean anything when added together.  This might make more sense to me if we were actually measuring the contributions of the various facets in wins, which is something that they actually do contribute to in a joint fashion.

10+8+6 = 24.

***

And you are right, we *shouldn’t* care!  This whole “percentage” thing is ridiculous, because we don’t need to use it for anything.

The allocation of payroll is driven by WAR, and as it turns out, the total number of WAR of your pitchers is roughly 43% of all WAR.  It is no surprise that pitchers in MLB also happen to get close to that percentage split in payroll.

Win Shares allocates roughly 35% of its win shares to pitchers, exactly because it STARTS with the percentage split.  And, as you are surmising, why should we even care about the split.

Well, I knew where the 24 came from, but I still don’t think we’ve defined what it represents.

As for WAR and payroll allocation, I see that as one use for having an accurate split percentage-wise.  Of course I think you could rather quickly come up with a list of reasons why this shouldn’t necessarily hold true. 

Curious, what does WAR say for the split between hitting and fielding for non-pitchers.  Do they also get to a point where offense < pitching+fielding?

"Curious, what does WAR say for the split between hitting and fielding for non-pitchers.”

That’s the crux.  There is NO replacement level hitters and replacement level fielders.  There are replacement level PLAYERS.

The calculation is:
hittingWAA
fldposWAA
WAR = hittingWAA + fldposWAA + replLevel

So, there is no split between hitting and fielding, anymore than there is a split between hitting and stealing, or drawing a walk and hitting a HR. 

That’s why we should never use the split as a starting or intermediate point.  The split will simply become a result of the equation.

Someone asked me to continue this discussion, and I will do so later today…

Let’s come up with a distribution of talent for nonpitchers and pitchers as the following:

SD nonpitchers
-1.5 30%
-0.5 25%
+0.5 20%
+1.5 15%
+2.5 10%

This means that 20% of nonpitchers have a SD that is +0.5 from the mean of nonpitchers.  The SD of this group of nonpitchers is 1.323.  The mean is 0.

This is what it looks like for pitchers:
SD pitchers
-1.125 30%
-0.375 25%
+0.375 20%
+1.125 15%
+1.875 10%

The SD = 0.992.  The mean is zero.

All data is purely for illustration.

All I did here was simply to tighten the range of talent among pitchers.  The spread in pitching talent (0.992) is 75% that of the spread in nonpitching talent (1.323).

Think of the SD as wins relative to average.

Let’s make the dollar value of the first rung of players as $0MM dollars for each group (the replacement players).  And then let’s make each SD of wins worth $2.5MM.

The average salary per pitcher is $2.8125MM.  The average salary per nonpitcher is $3.75MM.  The average salary of the pitcher is 75% of the nonpitcher.

All we did was convert the spread of standard deviations into dollars.  The unit of SD is wins.  And the unit conversion is dollars per win.  As you can see, the relative dollar value is identical to the relative standard deviation spread.  We do NOT want the variance, but the standard deviation.  The unit of variance is wins-squared.  We don’t care about wins-squared.  We care about wins.

***

If we presume 11 pitchers per team and 14 nonpitchers, then the total salary, using this illustration is 31MM for pitchers and 52.5MM for nonpitchers, or 37% going to pitchers.  That 37% of salary and WAR.

In reality, 43% of salaries are going to pitchers.  GuyM, I believe, (or perhaps it was Rally), had the foresight to suggest that since pitching had an uncertainty level, that teams were paying more for win.  He has used as an example that if the riskless spread had said that pitching should get 43% of the value, then pitchers should get 50% of the salaries, the extra premium for the extra risk.  His idea still holds here.  I’m saying, using this illustration, that the spread in talent suggests that pitchers get 37% of the WAR and would get 37% of the salary.  And then, using the risk premium, should get bumped to 43%, which is where it is.

***

Now, I presumed the spread of pitching was 75% that of nonpitching.  If instead we presume that the spread in pitching is 85%, then 40% of the salary (and value and total WAR) will go to pitching.

Tom,

Thanks for picking this thread up.  Like I said, I was really hoping to see how this one turned out.  For others that maybe interested, here’s the snippet that was posted by Tom over at baseball-fever.com that sparked my renewed interest…

Here’s where it gets tricky. Let’s stick to team-level for the moment, and then we can discuss individual players tomorrow. I’ll just some numbers for illustration purposes only, but they’ll be educated guess.

The following is the standard deviation of runs per game:
off: 1 SD = .60
def: 1 SD = .60

On this basis, and this basis alone, we can say that offense and defense is split 50/50. We can do this for every sport, and, I would bet, all sports have a similar split.

Let’s also say that:
off: 1 SD = .60
pitching: 1 SD = .48
fielding: 1 SD = .36

Let’s just take it for granted.

We know that it works out because:
.60^2 = .48^2 + .36^2

(That’s the 3,4,5 triangle for you guys wondering if Pythagoreus would ever apply in the real world.)

So, as I said, I work off the standard deviation to spread the wins. The “problem” is that if you only look at two component, off gets 50%, and def gets 50%. But, if you look at 3 components, the 60/48/36 split will give more shares to pit+def.

Alternatively, you can take the 50% split that the defense has and split it 48/36 between pitching and fielding (basically 28.5 pitching, 21.5 fielding).

So, that’s the basic framework.

All numbers above for illustration purposes only.

Things will change based on number of Ks per PA.

And things get more complicated when looking at each fielding position.

These points are for another day. Are we on the same page otherwise?

I guess what I was hoping to resolve was how we move from the 50/50 offense/defense split to the split of offense/picthing/fielding where offense then receives < 50% of the credit.  We’ve had a number of examples now with a variety of units wOBA, runs, wins.  I think it would be beneficial to settle on one and work from there.

I’m not sure exactly what the issue is.

When you treat offense, pitching, and fielding as three independent prongs, the split is roughly 5:4:3.

When you treat pitching and fielding as one unit, the split between offense and defense is 50/50.

When you treat offense and fielding as one dependent unit, the split between nonpitchers and pitchers is 60/40.

The splits is determined by the standard deviation, not variance, which was the point of my last post.  The unit of measurement of standard deviation is runs or wins.  The unit of measurement of variance is runs-squared or wins-squared.

The allocation of talent is based on money, which has a unit of measurement of dollars.  And dollars has a linear relationship to runs or wins.

So, while the variances add, as a property of variance, this is irrelevant in determining the split in allocation.

Until we can accept that things won’t add up the way we would like things to add up, we’ll be at an impasse.

I’ll try one last time.  If we determined that this equation is true:
1 SD = 10 runs, offense
1 SD = 10 runs, pitching
1 SD = 10 runs, fielding

We would therefore conclude that, be it standard deviation or variances, that the split should be 33/33/33.

But, if I asked how much of the money should go to nonpitchers and how much to pitchers, if you used variances, you would say 66.7/33.3.

However, if you do this:
nonpitchers = sqrt(10^2+10^2) = 14.1
pitchers = 10
splis = 59/41

Now, if you had a team of 9 hitters and 9 fielders, meaning that all three components of hitting, fielding, and pitching were independent, then, certainly, you would allocate only 33% of your money on pitching.

However, we have 9 nonpitchers, and the great hitters may have bad fielders or average fielders or good fielders.  The net effect is that the spread in talent for nonpitchers, in this illustration is 14 runs.

If you use the variances to apportion the money (or value), then you would be allocating the exact same amount to pitchers, regardless whether you have an MLB setup of two-way players or NFL setup of one-way plaeyrs.  And that makes no sense.

Tom,

I really appreciate your willngness to pick up this thread.  Although given the lack of other responses, I’m wondering if this may just be a conversation between you and I at this point.

Anyways, my primary interest here is in what the spread of talent (and how you are measuring it) suggests about the proper distribution of credit in something like WinShares or a similar stat.  I expect that the allocation of dollars would and should be dependent on that breakdown and is merely a by-product.

So I guess I’m just concerned that there is an inherent flaw about the assumption we’re making about the way these things should be added, if on one hand it suggests that I should be allocating 50% of the credit to my offense with 2 variables, and then less than 50% if I then add another variable to the mix, especially when we know that variable has absolutely nothing to do with the offensive part of the game.

I apologize if you feel like you’re beating a dead horse.  If that’s the case, let me know and I’ll just be content to dwell on this one myself.

No, it’s fine.

The question is not about adding an extra parameter.  But, about about an extra DEPENDENT parameter.  It’s not like you are adding Park as a variable.  You are adding the defensive portion of a player already in your equation.

Whatever model you construct must be able to distinguish between an MLB model and an NFL model (non-two way players).

Let’s talk about the NFL, and presume you have an off team, a def team, and a specialty kick team.  Whether you include the specialty kick team or not will NOT change the relationship between the off and def team.  That relationship will remain, because the specialty kick team has nothing to do with one or the other.  (In reality, it probably has because of field position, but I’m trying to come up with a decent illustration.)

So, you must stop thinking of it as “adding one parameter”, as if we are not distinguishing between NFL and MLB style of player distribution.

I guess to me it doesn’t much matter whether we’re talking baseball or football.  In either case though, I’m thinking that the only thing that each of the units commonly contributes to is wins.  But if you already have a breakdown of the contribution of each unit by wins, then you already know the answer to the contribution question, right?

I’ll refer to posts #46-49 for a previous example (which did not use wins) and one of my previous dilemmas.

If you are not seeing the distinction between MLB and NFL type rules, then I understand why you have a dilemna with my presentation.

Suppose you have a baseball league where hitting and baserunning were equally important, and there are no fielders, just pitching.  One SD is 10 runs in hitting and one SD is 10 runs in baserunning.

However, all you are told is that one SD in OFFENSE is 14 runs, which is the same spread in defense.  By this measure, you agree that you split the credit 50/50 between offense and defense.

And, once you are told that offense is equally hitting and baserunning, you split that 50/50.  So, you have 25 for hitting, 25 for baserunning and 50 for pitching.

But, the spread in talent is NOT twice as much in pitching that it is in hitting.  That spread is 40% wider.

***

Take another look: suppose that all the great hitters are kinda bad runners and all the bad hitters are kinda great runners.  One SD is 10 runs in hitting and one SD is 5 runs in baserunning. 

However, I just told you they are negatively correlated.  What you do with this information?

***

The answer is that you want to know the spread in wins, runs or dollars.  And that spread can be represented by standard deviations.  Or it can be represented by the absolute difference between the talent level and the mean.

In no way shape or form do you want to square those differences to get the value relationship.

It’s for this reason that all this “splitting absolute value” doesn’t work.

I’m pretty sure that I do see the distinction between MLB and NFL tpe rules.  So 2 questions…

1) If we’re talking about individuals, shouldn’t we be talking about an opportunity-based stat to measure talent, as opposed to a counting stat like runs?

2) If we take this conversation up to the team level (where I suspect the opportunity is not as relevant), does the process still apply?

Suppose you have a baseball league where hitting and baserunning were equally important, and there are no fielders, just pitching.  One SD is 10 runs in hitting and one SD is 10 runs in baserunning.

However, all you are told is that one SD in OFFENSE is 14 runs, which is the same spread in defense.  By this measure, you agree that you split the credit 50/50 between offense and defense.

And, once you are told that offense is equally hitting and baserunning, you split that 50/50.  So, you have 25 for hitting, 25 for baserunning and 50 for pitching.

But, the spread in talent is NOT twice as much in pitching that it is in hitting.  That spread is 40% wider.

I guess it’s not clear what you are proposing as the final answer for distribution of credit.  Are you suggesting that the 50/25/25 split of pitching/hitting/running is wrong in this scenario?

I’m using runs for ease of explanation.

And yes, the split is wrong if you are thinking that the split will give twice as much value to the total for pitching compared to the total for hitting.

***

Let’s try a practical example.  Here’s an 11-team league:
tot pit fld
89 80 9
25 40 -15
50 20 30
-14 16 -30
72 12 60
-60 0 -60
-24 -12 -12
-16 -16 0
-8 -20 12
-49 -40 -9
-65 -80 15

Tot is pit+fld.  The SD of tot is 50.0, pit is 40.0, and fld is 30.0.  The correlation between pit and fld is 0.

How much money are you going to pay for pitching and fielding?  Presume you’ve established you have to pay an average of 40MM for defense.

Well, first off, I’d suggest that i would need to know the relationship of runs to wins, since I want to pay for wins and not runs.  Then there’s probably some marginal utility that I’m comparing to since we’re talking money.

However, if we ignore the margin and assume a straight line relation from runs to wins, then the pitching portion will be no less than 57% 40/(40+30) and no more than 64% 40^2/(40^2+30^2).  I’ll admit that I don’t know which is right.

That’s the million $ question, and the reason that we are here!

Right, presume a 1:1 relationship between runs and wins.  Presume whatever marginal utility you want.

I’d like whoever is on the fence here to go through each team, figure out how much money you are going to pay the team for its pitching and its fielding.  Add them up, and tell me the average payroll for pitching and average for fielding.

Here are your two choices:

1. Set the replacement level at -120 runs for pitching and -90 runs for fielding, figure the $ per runs value of 0.190, and you get this:

tot pit fld 120 90 $pit $fld
89 80 9 200 99 38 19
72 12 60 132 150 25 29
50 20 30 140 120 27 23
25 40 -15 160 75 30 14
-8 -20 12 100 102 19 19
-14 16 -30 136 60 26 11
-16 -16 0 104 90 20 17
-24 -12 -12 108 78 21 15
-49 -40 -9 80 81 15 15
-60 0 -60 120 30 23 6
-65 -80 15 40 105 8 20

These columns represent this:
tot: pit+fld
pit: pit runs
fld: fld runs
120: pit runs over replacement
90: fld runs over replacement
$pit: pitching dollars
$fld: fielding dollars

The SD is 7.6MM for pitching and 5.7MM for fielding.  That’s 4/7ths for pitching.  The average payroll is 23MM for pitching and 17MM for fielding (that’s also 4/7ths).

Or…

tot pit fld varP varF 14400 8100 $pit $fld
89 80 9 6400 81 20800 8181 37 15
72 12 60 144 3600 14544 11700 26 21
50 20 30 400 900 14800 9000 26 16
25 40 -15 1600 -225 16000 7875 28 14
-8 -20 12 -400 144 14000 8244 25 15
-14 16 -30 256 -900 14656 7200 26 13
-16 -16 0 -256 0 14144 8100 25 14
-24 -12 -12 -144 -144 14256 7956 25 14
-49 -40 -9 -1600 -81 12800 8019 23 14
-60 0 -60 0 -3600 14400 4500 26 8
-65 -80 15 -6400 225 8000 8325 14 15

The varP and varF is the pitching and fielding runs squared (and carrying the appropriate sign).

The 14400 column is the wins above replacement level (which is 120^2 for pitching).  The 8100 is wins above repl fielding (90^2).

The dollars per runsSquared is .001778.

The total for the pitching is 26MM and 14MM for fielding.  The standard deviation is now 5.0 and 2.8.  Both are 64% for pitching.  (Note, we just went from a standard deviation split of 4/3 to 16/9.)

If you take the average of the +80 run pitching team and the +40 run team, you get +60 runs, and that costs you 32.7MM.  That’s 7.1MM above the average pitching team.

The +60 run fielding team gets paid 6.4MM above average. This is the variance method.

The variance method simply doesn’t work.

Sorry Tom, I meant to reply sooner.  I really appreciate the amount of effort you spent following up on this thread for a likely audience of one.

As for your last post, I’m not sure I totally followed it after you introduced replacement levels and dollars.  It seems like your primary argument for choosing SD over variance is based on how things correspond to the current allocation of MLB budgets.  Is that a correct interpretation?

Not “current”, but “expected”.  There is a direct 1:1 relationship between standard deviation, value over replacement, and salary.

Are we all agreed that the relationship is based on the spread in talent based on the actual spread (standard deviation) and not the square of the spread (variance)?

Like I said, I may be an audience of one here.  So, if by “we” you mean you and I, then I think it’s safe to say that you’ve convinced at least 50% of us.

The main sticking point for me all along has been the fact that on one hand you say that the game is split 50/50 between offense and defense and then later say that in another split pitching+fielding accounts for more than offense alone. This has pretty logical problems at the surface level.  I do agree that, in a theoretically perfect world, there would be a 1-to-1 relation between the allocation of win impact and $.  I’m just not sure that what we’re doing is truly measuring win impact.  For that I would be more inclined to look at an uber stat like SLWTS or Win Shares.