You're here at Beyond the Box Score, and that probably means you've got some interest in baseball statistics and/or sabermetrics. And if you're interested in those things, it stands to reason that you have some familiarity with new(er) metrics that try to establish overall value for a player by determining how many runs (and / or wins) a player is worth. That's the functional desire of metrics such as wins above replacement.
Well, if you're not deep into the math behind how these numbers are calculated, you may only have a cursory knowledge of how the conversion of runs to wins works. Historically, there's been a quick heuristic that you could use to translate runs into wins -- one that's still published over at FanGraphs in their library. One win is equal to (about) 10 runs, and vice versa.
So, I had the idea to examine the changing run environment and how runs are at more of a premium these days than in the past. And as something of a wins above replacement investigator, I wanted to see how that was affecting WAR values.
What I wound up doing was re-tracing the steps that originally brought us to that 10 runs = one win conversion, something I'm sure several other very smart sabermetricians have done before, for the 2013 season -- a season that brought us lots of strikeouts and not-so-lots of runs. And in doing so, I think I found that our old heuristic no longer applies.
Let me stress this first: this isn't a breakthrough -- it's something that's certainly known by smart sabermetrics experts and plenty of folks already know and apply it. But for those of us out there in what's not-so-affectionately known as the "liberal arts wing" of sabermetrics, it's not exactly common knowledge. I've often defaulted to the 10-to-1 rule when doing back-of-the-napkin calculations.
I simply want to walk you through my process and findings, and maybe that'll help us make fewer mistakes and work with slightly better information.
Before we get too much deeper, fair warning -- this is kind of a mathematics-heavy post, which is admittedly not my forte. So if you spot an error in my methodology, please stop by and let me know in the comments.
Let's start with the basics.
As it understand it, this heuristic -- that 10 runs equal a single win -- was calculated and based on the idea that we can find a team's expected winning percentage, then figure out how many runs scored / runs allowed a team needs to add or subtract in order to change a team's expected number of wins by one.
Since I assumed that with fewer runs being scored throughout the league, it takes fewer runs to change a team's expected winning percentage, I tasked myself with re-doing the math, specifically for last season's run environment. And in order to do so, I had to learn a little more about expected win percentages, and Pythagenpat.
So how do we find a team's expected winning percentage? We start with the most widely-used formula, unsurprisingly popularized by Bill James. It's the Pythagorean expectation formula. Pythag -- for short -- uses the following formula to determine a team's expected winning percentage:
runs scored ^ 2 / runs scored ^ 2 + runs allowed ^ 2
It's easy, it's simple, and it's not good enough* for what we're trying to do here. See those exponents of two? They're not very exact. By modifying those exponents based on what I understand to be good math (read: math I didn't do and only barely understand), we can do better.
* - Thanks to Bryan Cole and John Choiniere for helping place me on the right path.
Pythagenpat, developed by Patriot and David Smyth, is a refined, improved version of the formula*. Instead of using the exponent of two, it replaces the exponent with one that is more accurate, and also adjusted for the run environment.
* -- Pythagenpat is also not the only formula that does this. There's the Tango Distribution, and others. I picked Pythagenpat because of its wide use and perceived accuracy.
Since the run environment matters to us (it's WHY we're exploring this in the first place), it seems as if it is more responsible to use this formula:
runs scored ^ X / runs scored ^ X + runs allowed ^ X
Okay, cool. That variable X is the Pythagenpat exponent, which we need to figure out before we can go any further. The formula for that is:
runs per game ^ .287
Fabulous. We can actually start putting numbers in place of variables now.
In 2013, in all of MLB, 20255 runs were scored during 2431 regular-season games. Doing the math, that gives us an average of 8.332 runs per game. When we take that to the exponent provided, we get our Pythagenpat exponent of 1.837615.
Now we can actually run Pythagenpat for the season, using 1.837615 in place of X.
runs scored ^ 1.837615 / runs scored ^ 1.837615 + runs allowed ^ 1.837615
Looks beautiful, doesn't it?
I start by giving a team the league-average amount of runs scored and runs allowed in 2013. Since we're going by league-averages, these two numbers were the same. After all, the same number of runs were scored and allowed. I mentioned before that 20255 runs were registered during the 2013 season, so that means if that were averaged over all 30 teams, an average team would have scored 674.89 runs and allowed 674.89 runs, and that would (hypothetically) allow them to be a .500 team.
To make sure I'm not taking crazy pills, I plugged those runs scored and runs allowed numbers into our Pythagenpat formula:
(674.89 ^ 1.837615) / ( (674.89 ^ 1.837615) + (674.89 ^ 1.837615) )
Math ensues, and I get a winning percentage of, exactly, .500. When I multiply that winning percentage by 162, the number of games in the regular season, I get 81 wins. Perfect, that's what I was hoping for.
For a sanity check, I also ran Pythagenpat for a 2013 team that had a very, very similar number of runs scored and runs allowed -- the Los Angeles Angels of Anaheim. Their 733 runs scored (above the league average) and 737 runs allowed (again, above the league average) gave me a projected winning percentage of 0.4975, which comes out to a little over 80.5 wins, which rounds up to 81. I feel like my sanity has been checked.
The next step is to examine and see if adding ten runs to the calculation will still shift the needle, so to speak, by one win. So I took my league-average runs scored for my imaginary league-average 2013 team, and I added ten runs. Hypothetically, this should make my team one win better. I used this updated equation.
(684.89 ^ 1.837615) / ( (684.89 ^ 1.837615) + (674.89 ^ 1.837615) )
If the heuristic of 10 runs equaling one win were true, I should get a winning percentage of about .5061 or .5062 out of this formula. But I didn't. I actually got a winning percentage of .5068. And while that doesn't seem like a big difference, when you multiply that by 162 games, you get 82.09 wins instead of, say, just 82 wins.
10 runs doesn't equal one win -- 10 runs equals something closer to 1.1 wins.
My thought process at this point: if 10 runs are too many, let's try nine runs instead. Now we're adding nine runs to the hypothetical team's runs scored, instead of 10.
(683.89 ^ 1.837615) / ( (683.89 ^ 1.837615) + (674.89 ^ 1.837615) )
Doing THAT math gets me a winning percentage of .5061. Multiply that by 162 games and we get, believe it or not, 81.99 wins. Almost exactly one win.
It turns out that, given last year's run environment, nine runs are worth about one win, not ten.
Of course, I never trust my own math, so I went back and re-did the numbers in a couple of different ways*. Instead of adding the runs to runs scored, I added them to the runs allowed. The winning percentage decreased by the same amounts. I subtracted nine from the runs scored instead of adding it, same situation. Lather, rinse, repeat.
* -- All of that gyration might seem a little stupid to those of you with more math confidence / chops than me ... of course the numbers should read the same under each of these conditions. Still, I'd rather be careful.
So what now? Does this change everything? Do we tear down the established WAR frameworks and build something new in their place? Do we recognize Mike Trout as even more of a sabermetric darling*? Do we send nasty emails to Sean Forman and David Appelman demanding that the new rules be held up for all to see?
* -- Hah. We all know that's not even possible.
When sites like FanGraphs, Baseball-Reference, and Baseball Prospectus convert run values into win values, it doesn't appear as if they're using a simple 10-to-1 runs-to-win conversion, or anything like that. They already know this.
As a follow-on to this part of my research, I looked into how many runs above replacement a small sample of players were worth in 2013, as well as how many wins above replacement these players were worth. Take a look at the rates in the table below:
|FanGraphs RAR/WAR||Baseball-Reference RAR/WAR||Baseball Prospectus VORP*/WARP|
* - For position players, it's actually VORP + FRAA, as VORP doesn't take into account defense.
It's obvious that these sites are using some form of sliding formula. These rates are not consistent from year to year. These rates are not always consistent between players. The people in charge are obviously doing something to compensate for the changes in the run environment.
In addition, those variances between pitchers? They're probably meant to be a feature, not a bug. As is reported in the FanGraphs Library, the runs-to-wins conversion for pitchers changes based on how good a pitcher is. After all, during games in which, say, Felix Hernandez pitches, runs are at more of a premium. Since he gives up fewer runs, as a rule, it takes fewer runs to get to a win.
In the case of a pitcher like Edinson Volquez or Joe Saunders (in 2013, at least) those guys give up more runs, so it requires more runs to get to that win.
I am still surprised about some of these differences. 9.5 runs/win is actually kind of a big difference from 9.0 runs/win. 9.75 runs/win is even bigger. I'd figure that McCutchen would be the same or even less than Trout, given his league.
So that leads to another question: are these data providers / analysts using the right formula? I wish I could answer that with any degree of confidence, but I really can't right now. I don't know enough. But my (slightly) educated guess is this: we should be using something closer to 9-to-1 as the runs-to-wins conversion right now. FanGraphs' numbers seem more logical to me than Baseball Reference or Baseball Prospectus's numbers. But hey, I'm not a wizard, and judging things just on the way they seem is how Derek Jeter gets Gold Gloves.
Another thought: if it's not being done already, might we want to adjust the calculation -- slightly -- for league?
Here's the run difference between leagues for 2013:
|2013 Runs Allowed||2013 Runs Scored|
So, the National League's run environment looks to be slightly stingier than the American League, which makes logical sense, if for no other reason than the designated hitter. So runs appear to be slightly scarcer in the Senior Circuit. At the same time, with the constant interleague play, I wonder how much of a real difference that is. After all, the NL has a net deficit of runs to the AL.
Regardless, the takeaway for me -- and the takeaway that should be in place for many of the armchair analysts and sabermetricians out there -- is that 10 runs isn't a win. Nine runs is about a win, at least in this current run environment, and that probably could change depending on a few factors.
And the even bigger and better takeaway is, like I've mentioned in previous articles, it doesn't hurt to occasionally (1) reassess what you think you know, (2) try to figure something out for yourself, and (3) when something doesn't seem right, assume there's something else you don't know. Then repeat the process over and over again until your brain hurts and you just go for a run instead.
But for now, whenever I quickly think about how many runs go into a win, I'll think of nine instead of 10. At least, until the environment changes again.
. . .
Bryan Grosnick is the Managing Editor of Beyond the Box Score and a contributor to SB Nation MLB. You can follow him on Twitter at @bgrosnick.