Linux & Things: A Pretty Good Average

June 12, 2011: This post completely fraks up the calculation of David Smyth's Base Runs statistic. I've now fixed that, and added the data from 2009 and 2010. You can find all the updated tables here.

I'm a big fan of Sabermetrics, the use of statistical information to understand how baseball teams win games. Part of this is my love for the game, part my natural tilt toward numerical data, and part is that I've always enjoyed reading Bill James's work (full disclosure: he and I overlapped at KU, though we never met). Not to mention the fact that, from my desk, I can see several editions of both The Baseball Encyclopedia and Total Baseball.

But … in the old days, we judged batters by average (> 0.300 is good), home runs (> 30), and runs batted in (> 100). That was it. These stats have some problems: batting average doesn't tell you how many times a guy gets on base by walking, you can only bat in runs when your teammates are already on base, and as for home runs — well, OK, home runs are a pretty fair way to determine part of a players value.

The inadequacy of the traditional trio of AVG/HR/RBI led to the development of new measures for player performance: On-base percentage, slugging average, runs created, etc., etc. The problem is that off the top of my head I don't know what's a good number for any of these statistics. OK, a slugging percentage of 0.900 is better than 0.400, but is a player who slugs 0.500 a power hitter, or just Joe Shlabotnik?

This post is an attempt to get a handle on some of those questions. I've taken individual player data from 1995-2008 from Sean Lahman's Baseball Archive Database, selected those players who were eligible for batting titles under rule 10.22a (a minimum of 3.1 plate appearances per scheduled game), and determined averages and standard deviations for each category for each year in the database.

Limiting the study to post-lockout seasons means that we're looking at the modern game, i.e., small parks, large batters, lots of players attempting home runs, and consequently lots of players striking out. The restriction to players eligible for the batting title means that we're only looking a everyday players, i.e., those that some organization has deemed worthy be a starter on a major league team. This means that the average values we find will be somewhat higher than the true average over all major league players. That's OK, we're looking for a definition of good, not average.

Oh, one more thing. The database treats players that were traded during the season as two separate entities. So, for example, it has two entries for Manny Ramirez, Boston 2008, and Manny Ramirez, Los Angeles, 2008. Neither was eligible for the batting title, so the 2008 Manny isn't these averages.

To show you how it works, let's look at the traditional statistics from these years:

Year	# Players	Batting Avg.		Home Runs		Runs Batted In
		Ave.	σ	Ave.	σ	Ave.	σ
1995	125	0.286	0.028	17.83	10.53	73.40	24.04
1996	137	0.288	0.027	21.36	12.81	84.99	29.39
1997	134	0.283	0.029	19.97	11.29	80.96	26.26
1998	148	0.287	0.026	20.93	13.01	82.50	28.44
1999	152	0.290	0.026	22.02	12.61	85.40	28.49
2000	150	0.288	0.030	22.01	11.73	85.70	27.73
2001	146	0.282	0.028	21.62	13.40	81.89	29.11
2002	143	0.278	0.027	20.78	11.22	78.94	24.13
2003	146	0.281	0.027	19.86	11.06	78.86	24.17
2004	154	0.284	0.026	20.79	11.01	79.12	23.58
2005	143	0.280	0.022	19.97	10.94	78.04	23.31
2006	146	0.286	0.024	20.46	12.18	80.95	25.70
2007	156	0.283	0.028	19.13	9.92	79.76	23.37
2008	137	0.280	0.025	19.62	9.76	78.03	22.90
All	2016	0.284	0.027	20.49	11.61	80.70	25.99

So this tells us that since the year that must not be named, on average about 140 players played pretty much every day in a given year. They hit, on average, 0.284, with 20 home runs and 81 RBIs. The standard deviation for each statistic is given under the label σ, and for completeness we've added in the total for all regular-player-years since 1995.

OK, those numbers are a little bit lower than the ideal numbers I listed above (0.300/30/100), but remember we aren't just looking at a superstar, we're looking at all the guys who are good enough to play every day. And, really, if you didn't know anything else about a player, wouldn't you be inclined to take a guy whose stats were (0.284/20/81)?

But, as I noted before, Avg/HR/RBI isn't really a great determination of player worth. In recent years, the consensus has been that we're better off looking at how often a player gets on base (his on base percentage, or OBP), slugging percentage (SLG), and the sum of those two, On Base Plus Slugging, (OPS), where

OPS = OBP + SLG .

After all, if you can't get on base, you can't score, (and if you can't get on base, you'll only get 27 at bats in a game), and if you don't have some power, represented by SLG > AVG, then you won't drive in the runners that do get on. OPS, well, on its own OPS doesn't mean very much, but it is useful as a one-number statistic representing the worth of a player.

Here's the results for the players in our study:

Year	# Players	On Base		Slugging		OPS
		Ave.	σ	Ave.	σ	Ave.	σ
1995	125	0.360	0.036	0.460	0.075	0.819	0.100
1996	137	0.361	0.041	0.472	0.085	0.833	0.117
1997	134	0.354	0.041	0.460	0.075	0.815	0.108
1998	148	0.355	0.036	0.467	0.081	0.822	0.108
1999	152	0.364	0.039	0.479	0.077	0.843	0.107
2000	150	0.362	0.041	0.479	0.085	0.841	0.119
2001	146	0.353	0.040	0.470	0.092	0.823	0.127
2002	143	0.351	0.044	0.461	0.080	0.812	0.118
2003	146	0.351	0.039	0.461	0.075	0.812	0.108
2004	154	0.355	0.039	0.468	0.073	0.822	0.104
2005	143	0.348	0.034	0.458	0.069	0.806	0.095
2006	146	0.355	0.035	0.469	0.073	0.824	0.100
2007	156	0.354	0.036	0.458	0.066	0.812	0.095
2008	137	0.350	0.033	0.457	0.065	0.807	0.089
All	2016	0.355	0.038	0.466	0.077	0.821	0.107

That means, of course, that the average regular player gets on base 35.5% of the time, slugs just under 0.500, and OPS is over 0.800. It also means that anybody with an OBP/SLG/OPS line better than 0.355/0.466/0.821 is a pretty good player.

While AVG/HR/RBI and OPB/SLG/OPS tell a good deal about a players ability, that hasn't stopped people from generating other statistics. The idea that one number can represent a player's ability is particularly popular. The most famous of these, of course, is Bill James's Runs Created (RC) (we'll use the technical version). Another, similar statistic is David Smyth's Base Runs (BsR), which has the advantage that it will give the correct number of runs for extreme cases, such as a team that only hits home runs.

Now both Runs Created and Base Runs are cumulative stats, that is, the more games you play, the higher these stats get. To get a better idea of what these look like in a per game situation, I computed the number of outs made by a player, where

Outs = At Bats - Hits + Caught Stealing + Sacrifice Hits + Sacrifice Flies + Grounded into Double Play

and assumed that 24 outs made a game. So, for example, a player who made 480 outs in a season would have played 20 games, if he played by himself. So I'd divide his Runs Created and Base Runs by 20 to get RC/Game and BsR/Game, respectively. (For what it's worth, the players in this study averaged 413 outs per season, or 17 games worth of outs, at 24 outs/game.)

Year	# Players	Runs Created		RC/Game		Base Runs		BsR/Game
		Avg	σ	Avg	σ	Avg	σ	Avg	σ
1995	125	101.88	24.49	6.587	1.584	86.41	22.60	5.584	1.459
1996	137	116.08	30.93	6.739	1.838	99.16	28.23	5.767	1.741
1997	134	109.42	28.46	6.394	1.725	93.94	25.14	5.496	1.556
1998	148	113.96	27.06	6.535	1.620	96.94	26.19	5.556	1.543
1999	152	117.44	29.20	6.888	1.735	100.54	26.80	5.892	1.579
2000	150	116.83	31.93	6.851	1.964	100.61	28.39	5.909	1.786
2001	146	110.87	31.86	6.457	1.899	95.71	31.06	5.584	1.905
2002	143	106.99	28.27	6.251	1.972	93.73	27.35	5.499	1.978
2003	146	108.87	28.24	6.375	1.735	93.02	26.00	5.451	1.653
2004	154	111.54	27.71	6.516	1.908	95.62	25.71	5.606	1.939
2005	143	107.46	25.60	6.176	1.410	92.19	23.23	5.318	1.368
2006	146	113.09	25.25	6.542	1.432	96.13	24.18	5.570	1.455
2007	156	110.98	27.57	6.442	1.544	93.58	23.37	5.437	1.350
2008	137	108.99	25.17	6.284	1.462	92.95	21.49	5.367	1.291
All	2016	111.18	28.32	6.505	1.722	95.16	26.02	5.576	1.636