In between cheap jokes and a poor frame (alien invasions - how trite!), last time I occupied this space we discussed the real way to measure offense: by using runs. You can read about that here. The basic gist is that the only stats that make sense in baseball are the ones that measure contribution in runs. Runs are what help you win in baseball.

But how do you compare players to each other? Today I'm going to talk about baselines - to what standard should we hold players? And why?

Everything in life has a baseline. Think about your salary. Imagine that you make $40,000 per year, working at a comfortable but mind-numbing office job in which you spend the majority of your time posting on AN. This might be hard for some of you to imagine, but stick with me here.

There are a few ways you can measure your salary: in absolute terms (you make $40,000 every year), in average terms (you make $10,000 less than the average California salary), or in poverty terms (you make $30,000 more than the poverty threshold). Offensive production in baseball can be thought of in much the same way.

Before we go any further, remember from last time the only way to measure offense is in terms of runs contributed to a team offense. Also, we're going to adjust the numbers so that they are in terms of runs contributed to a team's offense over 690 plate appearances. (Why 690? There were 2268 games played in the AL last year, and in those games there were approximately 86,609 plate appearances. That means that a typical team got 86609/2268 = 38.2 plate appearances per game. Since there are nine guys in the lineup, a typical batter got 38.2/9 = 4.25 plate appearances per game, or around 690 plate appearances for 162 games.)

Let's pick a random full-timer from last year's team, say Nick Swisher. According to my last Staturday post, Swish created 97 runs in 660 plate appearances, or about 103 runs per 690 PA. Now we want to compare him to other players. How would we do that? Let's go through the three ways I described above:

1. In absolute terms
That's easy; Nick Swisher contributed 103 runs to our offense. The baseline here is zero. This is the true number of runs that Swisher contributed to our offense. But that doesn't really help us compare him to other players, since we don't know what the average player did.

2. Compared to average
This is useful, because average has a singular definition in baseball: an average team would win half of its games and lose half of its games. Last year, the average number of runs scored in an AL game was 4.9. So, it stands to reason that an average player would have created 4.9 runs in a game. From the arithmetic above, we know that there were on average 38.2 plate appearances in a game. So, the average player would have created 4.9/38.2 = 0.13 runs per plate appearance, or 88.5 runs per 690 plate appearances.  UPDATE: Bad wording, see monkeyball edit.

In other words, Nick Swisher created almost 14.5 runs more with his bat than an average player did. We all know that Nick Swisher is a very good hitter, probably one of the top 30 in the AL. Now we know that had Swish been replaced with an average hitter, the A's would have scored 14.5 fewer runs last year.

3. Compared to poverty line
When I say poverty, I mean baseball poverty. What if you made a team out of the worst players in the majors? You would likely have a very bad team. If you replaced all the guys on your team with junk that you can find lying around AAA, or last guys off the bench like Willie Bloomquist, you would probably win only 50 games. That is why this level of player is called "replacement level." The "R" in "VORP" and "WARP," for those of you who have ever wondered, stands for replacement level.

Why is replacement level important? Well, how much would you pay a team composed of replacement level talent? The minimum salary as allowed by the league. So, you can think of team-building this way: I pay the absolute minimum salary for my first 50 wins, and I have to spend at least that much money in order to field a full team. For any wins that I get beyond the first 50, I have to pony up some cash.

Replacement level is an economic concept, and it helps us think about how teams are built. The last guy on your bench, making the major league minimum, should be at least a replacement level player. Otherwise you are paying him too much money!

How do we compute what replacement level is? There's a nifty equation called the Pythagorean Equation of Baseball, which states that a team's winning percentage is:

WinPct = RS^2/(RS^2+RA^2)

WinPct is the fraction of games a team wins, RS is the average number of runs scored by the offense, and RA is the average number of runs allowed by the pitching/defense. If you're not sure what "^" means, it simply means "raise to the power of." That is, RS^2 means RS to the power of 2, or RS multiplied by itself.

(Aside: we can argue until we're blue in the face about the validity of this equation, but a) this is neither the time nor the place and b) you'd be wrong anyway and c) you'd cling to your prejudices so strongly that I could never convince you that you're wrong. If I sound arrogant about the Pythagorean Equation, it's because I've wasted more than enough time explaining to people why it's right when they think it's wrong.)

So what is the winning percentage of a replacement level team? It's simply 50/162 = .308. Let's call that .310 for simplicity. A team of average pitching and defense gives up 4.9 runs in a game (RA = 4.90). Now that we know WinPct = .310 and RA = 4.90, we can compute the number of runs a team must score in a game (RS) in order to be considered a "replacement level offense." A little arithmetic shows us that the answer is 3.3 runs. That is a very, very bad offense. Chicago, with the worst offense in the league last year managed to score 4.3 runs per game, so the ability to produce a team that has replacement level offense is very difficult - although our sister team across the bay is apparently trying to do just this.

As before, we compute the number of runs a replacement level hitter creates per plate appearance by dividing the number of runs by the average number of plate appearances in a game: 3.3/38.2 = 0.086. Or, per 690 plate appearances, 59 runs.

Thus, we can say that Nick Swisher created 103-59 = 44 runs over what a replacement level hitter would have created. That is, if we had replaced Nick Swisher with AAAA junk, we would have scored 44 fewer runs.

UPDATE: I think I'm wrong here.  A team full of replacement level players would not have only replacement level run scoring, they would have replacement level run prevention (pitching and defense).  So it's not really fair to say that a replacement level team would have average pitching and then compute the replacement level for hitters based off of that.  Other, smarter researchers have looked into this, so I am willing to cop to this mistake.  It turns out that offensive replacement level is something like .395, NOT .310. 

Why is that?  Well, a replacement TEAM will have a winning percentage of .310.  But both their hitters and pitchers will be of replacement quality.  Well, imagine that a team has average pitching but replacement hitting, and we set the replacement HITTING level at .395.  Then, using the equation above with WinPct = .395 and RA = 4.9, we solve for RS and find that RS = 3.96 for replacement hitting. 

Now repeat for a team of average hitters but replacement pitchers/defenders, and you'll find that RA = 6.06 for pitchers/defenders.

So a TEAM of replacement run scoring and replacement run prevention woul have RS = 3.96 and RA = 6.06, which implies a winning percentage of .300 (close enough to .310 for my tastes).

Using the mathematics below, we find that the replacement hitter is good for 3.96/38.2*690 = 71.5 runs, instead of the 59 I stated above.  Redoing the numbers below will be left as an exercise to the reader.

Now, it's possible for reasonable people to disagree on what constitutes replacement level. I think you could make a good argument for any winning percentage between .275 and .350, and you will see various definitions in various places. My replacement level of .310 is not to be taken as gospel by any means.

Baseball Prospectus is famous for introducing two stats that use a replacement level baseline: VORP and WARP. VORP uses a pretty good definition of replacement level, but WARP does not. In fact, it is a bad idea to use WARP for...well, anything. Maybe pre-WWII historical studies, but that's it. WARP considers replacement level to be along the lines of the 1899 Cleveland Spiders, who won only 20 games in a 154 game season. Such a bad team will probably never exist again in the majors.

Another baseline

That's not the end of the story. Obviously, some positions are more difficult to field than others, and some hitters are not capable of playing the harder positions. A guy like Jack Cust is a great hitter, but Cust can't play shortstop. Likewise, Marco Scutaro isn't a very good hitter, but we all know intuitively that he's not that bad when you consider the fact that he's a middle infielder.

That's why, after applying the replacement-level or average baseline adjustment, we need to adjust for position. As Tom Tango explains, "These are needed so that an average fielding 1B is not valued the same as an average fielding SS." Shortstop is the more important position, and an average fielder there is much more valuable than an average fielder at 1B. Think about it this way: an average fielding shortstop would be an above-average fielder at first base, so the pool from which we can choose a firstbaseman is that much bigger.

I won't go into too much detail here, but suffice to say that the above-linked post gives a pretty good approximation as to how we adjust the baseline for various positions.

Pos   Adjustment

C        +10 runs

SS      +5

CF      +5

2B       +0

3B       +0

LF        -5

RF       -5

1B       -10

DH      -15

Note that this has nothing to do with how well a player plays this position; it is only a bonus (or demerit) given to players who can field a particular position. After all, a good shortstop is hard to find!  Notice that the hardest defensive positions get the biggest bonuses and the easiest defensive positions get the biggest penalties.

 Finally, let's check out some 2008 A's, as projected by the THT Season Preview. These projections appear courtesy of The Hardball Times.

Player                        Runs/690 PA              PRAR/690 PA
Spartacust               111                                    37
Chavvy                       97                                   38
BART                         96                                    27
Dad, Jr.                      96                                   27
MaEl                          89                                     30
Emil Brown               86                                   22
Hannahan               85                                     26
T-BUCK!                   85                                    21
ChoCoBoCro          81                                    27
Rob Bowen              81                                   32
Donnie Murphy         75                                   21
Ryan Sweeney        68                                    14
Zooks                       67                                     18
PRAR = position-adjusted runs above replacement = runs - replacement level + position bonus

Example: Jack Cust: 111 - 59 + (-15) = 37

You can tweak these numbers until your heart's content by including things like park effects, which I will not go into here. This framework gets you most of the way to a statistic like VORP. That doesn't mean that you should be computing all of these numbers for yourself, but if you've ever wondered where all these "new-fangled stats" come from...well, here you go.

I do want to remind you that what we're assuming that these players will garner 690 plate appearances, or basically a full season. So Chavvy is projected to be 38 runs above replacement only if he plays all year. Obviously, that number will be less should his shoulder, elbow, ass, and hamstring are all injured in a terrible bowling accident.

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If you enjoyed this post and the type of numbers presented here, come by The Hardball Times website. We publish original research on baseball statistics using the same kind of analytical framework you see here, such as Win Shares Above Bench. We also publish articles on baseball history, economics, and current events.

If you enjoy what you see there, please consider purchasing our THT Preseason Preview. It has great preseason essays for all 30 teams, as well as projections for tons and tons of players. Our book sales help us to cover the cost of purchasing and presenting the data on our website, as well pay a nominal amount of money to our dedicated writers, editors, and webmasters. If you do wish to purchase our book, consider purchasing it directly from ACTA Sports. It is more expensive than Amazon, but a greater fraction of the proceeds go to the authors and editors. If you're a hardcore free-marketeer and want to purchase our book through Amazon, please consider donating a portion of the cost difference to THT by visiting our homepage and finding the "Make a Donation" button the bottom of our right sidebar.

[Note by salb918, 02/14/08 4:16 PM EST ] I'm seeing now that the fine bloggers at USS Mariner have written a similar post using run values instead of win values.  Go on over and check their primer out as well.

[Note by salb918, 02/14/08 4:22 PM EST ] If you don't like the tables, take it up with Blez and clockwerks.  I've spent too long trying to figure it out and having AN 3.0 delete my changes to waste any more time on it.

[Note by salb918, 02/16/08 9:38 AM EST ] One more thing about our preseason book.  Each chapter is written by a hardcore fan of that team - if we couldn't find one on the THT staff, we went out to the blogosphere, and in some cases, the SB Nation blogosphere.  That means Jeff Sullivan of Lookout Landing wrote the Mariners chapter and Larry Borowosky of Viva El Birdos wrote the Cardinal chapter. 

UPDATE: I just wanted to clarify that the analytical framework outlined in this post was pioneered by...well, I don't know, but it was a long time ago and it sure wasn't me.  As before, I can't take any credit for the ideas.  But also as before, I hope this is a useful and easy-to-understand primer.  The goal of this post, as well as my last one, is to make the "new-fangled" stats a little less mysterious and a little more accessible.