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April 8, 2005

Averages and Extremes

Gleaning Information from Performance Spikes

by Michael Rawdon

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Batting Average gets no love anymore.

Three decades of sabermetric analysis has diminished the once-proud Triple Crown stat through bites (on-base percentage) and nibbles (batting average on balls in play). It's enough to drive any self-respecting free-swinger to distraction.

But batting average is important. In 2004, hits accounted for more than 70% of the on-base events in on-base percentage across the majors. The first base of each hit accounted for more than 60% of total bases in the majors, a key to slugging average. A single is more valuable than a walk, because it carries runner-advancement potential the walk mostly lacks.

So a better understanding of batting average seems like useful knowledge to have.

Batting average is a skill. Being able to choose to swing at a ball, put it in play and successfully reach first base before being thrown out is an ability which some players possess at higher level than others. Of course, it's also a famously volatile stat. Being a skill, if in a single season a player shows a large deviation from his previously-established level of hitting, then you'd expect that the next year he'd likely see a regression to that established level, wouldn't you?

This was the theory behind my analysis of Pat Burrell's poor 2003 season. In that, I used constraints of looking only at players who saw a big drop in batting average while their patience and isolated power remained about the same. Why not look at a much larger pool of players, without that constraint, and see how they performed?

Using stats from the Baseball Archive, I looked at players who logged at least 300 plate appearances for three consecutive years (to try to filter out the noise of very small sample sizes). The first two years form the "baseline," while the third is the candidate's "spike" year (S). The spike year was considered if the player's batting average (AVG) in that year was above the higher of the AVG in the two baseline years, or was below the lower. I used translated batting averages for every year starting with 1946. This yielded 6550 data points.

For each spike year S I plotted it against their regression year (S+1), without putting a PA requirement on the S+1 season. The spike year was measured in the magnitude of the spike, in AVG points, while the regression year was measured in the magnitude of the change from the spike year. So if a player dropped 50 points of AVG in one year, but gained 30 of it back the next year, his S value is -50 and his S+1 value is +30.

Of our sample, 2237 players (34%) fell between the two baseline numbers and so have S values of 0 (zero). Everyone else had a spike--if only a very small one--outside their baseline range. The data points plot this chart:

Regression to the mean is a powerful thing. According to the trend line, players whose batting average dropped from their baseline regained about 60% of it the next year. Players whose batting average increased lost about 80% of their gains.

(For those keeping score at home, the single biggest one-year drop in the set is one of the Dave Robertses. In 1974 with San Diego, Roberts lost more than 100 points of AVG from his baseline, then regained more than 120 points the next year, albeit in just 132 PA. The guy at the other extreme is more heralded: Andres Galarraga made a run at a .400 average in 1993 with Colorado, gained about 90 points in BA then gave back about 2/3 of it the next year.)

What's going on here is a combination of not-so-mysterious things:

  • Luck. Sometimes a hitter has a few more balls drop in--or get caught--and has a good year. It doesn't really reflect any change in his actual skills, so the next year he tends to revert towards normal.

  • Actual skills improvement. Sometimes a player really does learn a new trick and gets better at hitting.

  • Actual skills decline. Age or injury leads to deteriorating skills and a lowered ability to collect hits.

The table suggests that luck is a very strong factor in the performance of batting average, which explains its volatility. But the indication that players with an upward spike tend to lose more of their gains than players with a downward spike recoup their losses suggests that the improvement/decline elements also play a role. In particular, we could infer that skills decline is a more likely occurrence than skills improvement. (This baseball game must be a tough one to master!)

Let's look more closely at the characteristics of the data set.

First, what really constitutes a "large" spike? It's one which is sufficiently rare to be interesting. Players experience 5- or 10-point changes in their batting average year-over-year all the time; indeed, it's unusual when a player doesn't see at least a small change.

The following tables break down the tails of the chart, showing population of the tail for a spike of a given size or greater, and the percentage of players in that tail who continued to improve after an upward spike, or decline after a downward spike. This gauges for a spike of a given magnitude how likely a player is to regress towards the mean the next year:



Min Down    % Further
Amt (S)     Decline (S+1)    #Players
-5              34.4%            2054
-10             31.1%            1668
-15             28.1%            1321
-20             24.7%            1041
-25             21.5%             808
-30             20.6%             613
-35             20.0%             450
-40             16.5%             328
-45             15.7%             230
-50             12.6%             151

Min Up      % Further
Amt (S)     Improve (S+1)    #Players
+5              20.6%            1655
+10             17.6%            1323
+15             16.6%            1024
+20             14.8%             777
+25             12.2%             572
+30             10.3%             416
+35             10.5%             293
+40              8.0%             199
+45              7.5%             132
+50              7.9%              88

The 30-point spikes seem like an interesting dividing line: About one player in five with a down spike of 30 points or more will continue to decline, while only 1 in 10 with a 30+ point spike upwards will continue to improve. 40- and 50-point spikes show an increasing trend in this regard, albeit with a much smaller sample set to work with. But these tables provide a continuum of performance trends which can be used to study players who displayed a batting average spike in a given year.

The population distribution suggests another avenue: The 6550 data points in our set work out to about 4.8 players per team-year. (In other words, in a season a team is likely to have just under five players on it who have collected at least 300 PA in each of the last two seasons.) Of these players, about 1 in 6 will have a 30-point spike (or more, and either up or down), about 1 in 12 will have a 40-point spike (and will be a subset of the first set), and about 1 in 27 will have a big 50-point spike.

So with the current 30 major-league teams, this means that a season will have about 152 players who meet the PA requirement. About 25 of them should have had 30-point spikes, 13 of them 40-point spikes, and four or five should have had big 50-point spikes.

So let's look at the 2003 season's batting-average spikes, and see how those players did in 2004:


PLAYER                  S Age     S (2003)   S+1 (2004)    S+1 PA

Jason Giambi/NYA          32        -69         -42          322
David Bell/PHI            30        -67         +92          603
Larry Walker/COL          36        -51         +13          316
Pat Burrell/PHI           26        -49         +44          534
Mark McLemore/SEA         38        -46         +12          295
Paul Konerko/CHA          27        -46         +36          435
Bernie Williams/NYA       34        -44         -1           651
Bobby Higginson/DET       32        -42         +9           531
Craig Counsell/ARI        32        -42         +12          545
Tony Womack/ARI-COL-CHN   33        -41         +86          606
Kevin Millar/BOS          31        -41         +19          588
Todd Zeile/NYA-MON        37        -40         +6           396
John Olerud/SEA           34        -38         -12          500
Ryan Klesko/SDN           32        -37         +37          480
Eric Young/MIL-SFN        36        -33         +28          402
Desi Relaford/KCA         29        -32         -24          430

Todd Helton/COL           29        +30         -21          683
Ben Molina/ANA            28        +31         -13          363
Albert Pujols/SLN         23        +33         -32          692
Kenny Lofton/PIT-CHN      36        +33         -21          313
Mark Grudzielanek/CHN     33        +34         -14          278
Richard Hidalgo/HOU       28        +34         -70          578
Matt Stairs/PIT           35        +39         -24          496
Mike Young/TEX            26        +42         +2           739
Jason Kendall/PIT         29        +45         -7           658
Javy Lopez/ATL            32        +64         -20          638
Melvin Mora/BAL           31        +66         +13          636
27 30+ spikes, 15 40+ spikes and five 50+ spikes, a little on the high side, but not far off. More down spikes than up spikes, as expected. Down-spike players were a little more likely to continue going down than up-spike players were likely to keep going up. The down spike players tended to be in their 30s, while the up-spike players tended to be in their (late) 20s.

This brings us to the matter of age. To come clean, there is an age bias in this study, because in order to qualify for the study a player has to have three years of semi-regular major-league service. The median age at which a player logs his first 300-PA season is 25, so at least half the players wouldn't even have a chance to be part of the study before age 27. By contrast, many old players will qualify near the end of their careers.

With that caveat, the following graphs show the age distribution of players in the data set with spikes of at least 30 points:

Despite the bias against young players due to the PA requirement, up-spike players are still a little younger than down-spike players, as a group. Ages 23 and 24 seem somewhat more likely to produce an up spike than a down spike, while ages 33 and 34 are the reverse. But otherwise there isn't a big difference between up spikes and down spikes based on age.

What about the impact of age on a player's regression in his S+1 year? These next two charts show the average regression by age of players with at least a 30-point spike, where the regression is expressed as a percentage of the spike. So if a player's up spike was +50 and his regression the next year was -30, then his regression percentage was 60%. As a cutoff, I excluded ages for which there were fewer than 10 players (which I admit is somewhat arbitrary). For visual consistency between the two charts, the regression from up-spikes is expressed as a negative percentage.

Here I think we've found something: the older a player is, the less likely he is to bounce back from a big down spike, and the more likely he is to come back to Earth after a big up spike. In particular, once a player enters his 30s, a big down spike may indicate a significant skills erosion--he's likely to regain only about half of his losses--while he's also unlikely to maintain more than a small fraction of gains from a big up spike.

As I said earlier, many of these behaviors are not very surprising: We expect hitting statistics to regress to the mean, and for older players to struggle more than younger players But it's also valuable to have an understanding of how likely a player is to regress, and by how much.

With that in mind, I'll close out with your class of 2004 30-point batting average spikes:


PLAYER                    S Age     S (2004)
Chipper Jones/ATL           32         -62
Scott Spiezio/SEA           31         -52
Jacque Jones/MIN            29         -52
Geoff Blum/TBA              31         -44
Jason Giambi/NYA            33         -42
AJ Pierzynski/SFN           27         -39
Juan Encarnacion/LAN-FLA    28         -36
Bret Boone/SEA              35         -35
Luis Gonzalez/ARI           36         -34
Cliff Floyd/NYN             31         -33
Sammy Sosa/CHN              35         -33
Doug Mientkiewicz/MIN-BOS   30         -32
Jay Payton/SDN              31         -31
Edgar Renteria/SLN          28         -30
Timo Perez/CHA              29         -30

Juan Uribe/CHA              24         +32
Cesar Izturis/LAN           24         +33
Tony Womack/SLN             34         +36
Sean Casey/CIN              29         +38
Aramis Ramirez/CHN          26         +38
Carlos Guillen/DET          28         +41
Ichiro!/SEA                 30         +44
JT Snow/SFN                 36         +48
Jack Wilson/PIT             26         +51
Terrence Long/SDN           28         +51
Adrian Beltre/LAN           25         +70
Brandon Inge/DET            27         +77

Inge and Beltre rank seventh and 10th in up spikes since 1946 by my numbers. I don't expect them to repeat their lofty performances in 2005. Beltre was one of the big free-agent signings this past off-season, and even if he retains half of his spike his AVG will fall from over .330 to under .300 (his career high before 2004 was .290). That probably isn't what the Mariners thought they were paying for when they signed him.

Caveat Emptor.

Michael Rawdon is a programmer and happy Red Sox fan who lives in Silicon Valley, CA. His cats have remained strangely indifferent to the Red Sox championship, possibly because they're from Wisconsin. He can be reached here.

Related Content:  Regression,  The Who,  Year Of The Injury,  Player Age,  Batting

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