Runs Created by Decade: An Analysis in Variance

by GOOCH24 on Feb 10, 2008 6:31 PM EST Comment 31 comments

In the discussion over Albert Pujols's status as a Hall of Famer, a statistical debate arose over how we should compare players from by-gone decades in terms of OPS+. Valentin pointed out that a comparison to the average player (what OPS+ does) might be biased if there's a significant difference in the variance in OPS in past eras. IOW, Valentin suggested that it would matter if, say, a player who posted a 130+ in the 1950's was an extreme abnormality while a player who posted a 130+ in the 1990's was relatively common.

Well, this presents an empirical question: is there a substantial difference in the variance in offensive production in previous eras. To help answer this question, I have examined the Runs Created for MLB from 1950 to the present. Specifically, I've looked at the average Runs Created for each decade from 1950 to 2005 (latest year available), to see if there has been a substantial difference in the variation in runs created over those decades.

I employed the dataset on batting provided in Sean Lahman's Baseball Database. I excluded all players with less that 150 at bats in a season. The following data include the mean, median, & modes, the variance and standard deviations for Runs Created. I've used the following runs created formula:

rc_X1 = (H + BB - CS + HBP - GIDP);
rc_X2 = TB + (.26 * (BB - IBB + HBP));
rc_X3 =(.52 * (SH + SF + SB));
rc_X4 = AB + BB + HBP + SH +SF;

rc = rc_X1 * (rc_X2 + rc_X3) / rc_X4;

I've also included a quantile breakdown for runs created in each of the decades:

The Fifties (1950-1959)

N 1813
Mean 55.9209845
Std D 31.5128001
Var 993.056569
Med 49.45367

Quantile Estimate
100% Max 187.62018
99% 148.14395
95% 114.88000
90% 100.00556
75% Q3 76.41856
50% Median 49.45367
25% Q1 30.34681
10% 20.05333
5% 16.39892
1% 11.37026
0% Min 8.57665

The Sixties (1960 - 1969)
N 2289
Mean 52.29305
Std Dev 29.53063
Median 47.23479
Variance 872.05787
Range 169.64884
Quantile Estimate
100% Max 177.66250
99% 132.01471
95% 107.21445
90% 94.02014
75% Q3 70.89245
50% Median 47.23479
25% Q1 27.71607
10% 18.39132
5% 14.82995
1% 10.43723
0% Min 8.01366

The Seventies (1970 - 1979)

N 2918
Mean 52.75782
Std Dev 28.66373
Median 47.93912
Var 821.60919
Range 154.24352
Quantile Estimate
100% Max 160.26340
99% 129.96813
95% 103.82451
90% 92.60296
75% Q3 72.36598
50% Median 47.93912
25% Q1 28.57147
10% 19.13247
5% 15.67842
1% 10.78744
0% Min 6.01988

The Eighties (1980 -1989)
N 3162
Mean 51.70840
Std Dev 28.05438
Median 46.96388
Var 787.04827
Range 147.99317
Quantile Estimate
100% Max 153.60612
99% 126.75796
95% 105.90848
90% 90.73343
75% Q3 70.23333
50% Median 46.96388
25% Q1 28.43210
10% 19.38759
5% 16.01681
1% 11.61000
0% Min 5.61295

The Nineties (1990 - 1999)
N 3395
Mean 56.06562
Std Dev 31.74681
Median 49.65593
Var 1008
Range 186.91802
Quantile Estimate

100% Max 193.33921
99% 146.74168
95% 115.88431
90% 101.28195
75% Q3 76.41611
50% Median 49.65593
25% Q1 30.25041
10% 20.69598
5% 16.86006
1% 12.17636
0% Min 6.42118

The Oughts (2000 - 2005)
N 2639
Mean 59.70025
Std Dev 34.02325
Median 53.48486
Var 1158
Range 223.95212

Quantile Estimate

100% Max 230.40970
99% 154.90486
95% 124.12382
90% 108.89175
75% Q3 80.00087
50% Median 53.48486
25% Q1 31.56901
10% 21.79130
5% 18.15000
1% 12.58929
0% Min 6.45758

============
OK. So what do all those numbers mean? Well, for one, it is apparent that there has been some shifts in the variance in runs created over time. The maximum RC's have increased in some eras relative to other eras (Max RC in the Oughts: 230.40, Max RC in the 1980's: 153.60). This is consistent with the notion that the runs created distribution has expanded over the last few decades.

However, there doesn't seem to be a substantial difference in the standard deviations in each decade. IOW, while we've seen an expansion of the distribution at the tails, for the most part, the standard deviations in the bulk of the distribution remain relatively the same. We've seen a slight increase in standard deviations, relative to other decades:

1950's 31.51 runs
1960's 29.53 runs
1970's 28.66 runs
1980's 28.05 runs
1990's 31.75 runs
2000's 34.02 runs

So the max variance in comparing any one decade to another is 5 runs created in a season. So even if we weighted our OPS+ calculations to account for this difference in SD, I doubt we'd get much of a difference in our results.

I invite comment on my baseline assumptions, the results, and the conclusions I've drawn. D.GOOCH

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In a nutshell...

...what I would argue the data shows is that, while there has been an apparent right-ward skew in the data (hard to go the other way since the data is censored on the left), it is at best a slight increase in skew and runs created in each decade approximate a normal distribution. I see little evidence to suggest a substantial shift in the density of runs created such that it would justify a correction factor. D.GOOCH

Cardinals, Cardinals, Cardinals!

by GOOCH24 on Feb 10, 2008 6:48 PM EST reply actions 0 recs

So in terms

of the increase offense as it realates to the findings in.

http://steroids-and-baseball.com/

Isn't what they say match what you say.

The "steroid" era is a myth.

And if not.

Why?

by Harknights on Feb 10, 2008 6:56 PM EST reply actions 0 recs

On the steroids tangent...

Well, I think we have seen some increase in RC over the decades. The difference between 150 & 230 as an upper-end is not trivial. However, I'm in the "steroid era severely exaggerated" camp. I'd be more inclined to attribute the increase in offense to smaller parks, better and year-long training programs, the lowering of the mound, etc. etc. than I would to steroids specifically. Now, I can't rule out some steroids effect. But I do think to the extent that it exists it is fairly small.

But you should also note that it is possible that steroids had a substantial influence on hitters AND pitchers. So while there may have been significant effects, they tend to cancel one another out. D.GOOCH

Cardinals, Cardinals, Cardinals!

by GOOCH24 on Feb 10, 2008 7:32 PM EST up reply actions 0 recs

don't forget expansion...

... and the declining popularity of baseball relative to other sports. in other words, the talent pool may have gotten shallower.

now, perhaps, the dramatic increase in the number of Latin American players has diminished the second effect (i.e. less popularity), but i doubt that it makes up for the fact that football and basketball have so dramatically outgrown baseball on the youth level, for several decades now.

so, the truly talented can excel and pull the mean up on their own. we've seen this over the past decade or so; whether it's due to steroids or other factors (or some combination) is open to debate, of course, but we've still see an offensive increase in the tails, as you've described.

by kindred on Feb 11, 2008 7:18 AM EST up reply actions 0 recs

On a somewhat related subject

if Albert Pujols has 1,100 at-bats in 2008 and 2009 combined and collects 392 hits, he can claim this decades batting crown.

Still looking for 1985 Regular Season games on DVD/VHS

by Hardcore Legend on Feb 10, 2008 7:00 PM EST reply actions 0 recs

Differences in variance

Um, how can you say the variance isn't substantially different? I see a huge difference from the '80s to the '00s.

1980s 2000s N 3162 2639 Mean 51.7 59.7 Std Dev 28.1 34.0 Median 47.0 53.5 Var 787.0 1158.0 Range 148.0 224.0

The St. Louis Cardinals- 11 time World Champions!

by Zubin on Feb 10, 2008 10:39 PM EST reply actions 0 recs

An answer to your question

As I mentioned, there are differences. We do see an increase in the upper end of run creation. So the standard deviation is larger...I'm just not convinced it is sufficiently large enough to make for a case that OPS+ can't be compared across time.

Remember, OPS+ is calculated in relation to the mean. So a substantial increase in the mean isn't particularly important. The question for us is the shape of the distribution.

The 'problem' with RC (or most any offensive production stat for that matter) in terms of a substantial increase in the mean is that offense is left censored (with most offensive stats you can't have negative offensive production).

So we could invision a RC distribution that would approximate the poission distribution. As the mean increases, it would approach a standard normal distribution (as the center moves further away from the censor point).

But from what I can see from the distribution of RC scores, the eighties and the oughts both approximate normal distributions. Note the relationship between the median and the means. The exact same slight right skew. So the left censor doesn't really come in to play yere.

So while agree that run creation has inflated in the oughts in comparision to the eighties, I think the shape of the distributions are relatively the same. Hence we shouldn't have a concern regarding comparisions of OPS+. Consider the below hypothetical distributions

DISTRIBUTION 1
0
0
0
5
5
5
10
10
15
15
15
30
30
30

DISTRIBUTION 2
5
5
5
15
15
15
20
20
40
40
40

Note that the means and ranges of these two distributions are distinct. The mean for distribution 1 is 12.14 and the mean for distribution 2 is 18.21. The range for D1 is 30 and the range for D2 is 35. But the shape of these distributions is relatively the same. So a '20' in the second distribution, relatively speaking, means the same as a '10' in the first.

Valentin's concern regarding OPS+ was if the distributions were substantially different. Consider the following:

DISTRIBUTION 3
0
0
0
5
5
5
10
10
15
15
15
20
20
20

DISTRIBUTION 4
30
30
40
40
40
40
40
50
60
60
60

Distribution 3 here approximates a normal distribution. Distribution 4 is a flat distribution. It is decidedly NOT normal. So if there were substantial differences in the shape of the distributions of runs created in the 1980's in relations to the 2000's...well, I'd have some concerns regarding comparisons. As it is, I think the real world is closer to D's 1 & 2 rather than anything like D's 3 & 4. D.GOOCH

Cardinals, Cardinals, Cardinals!

by GOOCH24 on Feb 10, 2008 11:42 PM EST up reply actions 0 recs

Correction on Distribution 2

Whoops. Forgot the 10's:

5
5
5
10
10
10
15
15
20
20
20
40
40
40

Cardinals, Cardinals, Cardinals!

by GOOCH24 on Feb 10, 2008 11:47 PM EST up reply actions 0 recs

Yes, but Valentin and I were concerned

with the upper end of the distributions. When you compare Mattingly's and Pujols' best years, you are looking at guys in the upper 1% of the curve.

The St. Louis Cardinals- 11 time World Champions!

by Zubin on Feb 11, 2008 10:50 AM EST up reply actions 0 recs

Comparisons...

What Gooch is showing you is that since there is very little difference in distribution or standard deviation between different eras, comparing the OPS+ numbers for those players should give you a good comparison.

It doesn't matter that they are on that upper 1% of the curve or not; OPS+ adjusts for relation to the mean of the particular year. Since the standard deviation and distribution numbers are not much different from era to era, OPS+ for the upper 1% of players will not differ any more than the OPS+ of an average player. Hence, statistically it is a good tool for comparison between eras.

"I just wish that the late Harry Caray were still around so I could hear him mispronounce 'Kosuke Fukudome' every fukun' night" -- Dennis Miller

by fourstick on Feb 11, 2008 12:47 PM EST up reply actions 0 recs

Um, yes it does...

"while we've seen an expansion of the distribution at the tails, for the most part, the standard deviations in the bulk of the distribution remain relatively the same."

We are talking about two guys in the tails, not in the bulk.

GOOCH:
Any chance you'd be willing to put up histograms of the distributions?

The St. Louis Cardinals- 11 time World Champions!

by Zubin on Feb 11, 2008 2:50 PM EST up reply actions 0 recs

your wish...

I've posted a text file with the histograms and normal probability plots for each of the decades here:

http://www.donaldgooch.com/RC_HIST.txt

I don't have time to comment right now (off to class) but feel free to preuse and comment in the meantime. :)

Cardinals, Cardinals, Cardinals!

by GOOCH24 on Feb 11, 2008 3:30 PM EST up reply actions 0 recs

ok...

OK, I've got a little time to address this now. In taking a close look at the histograms, I think what I've said previously is apparent...perhaps more so than I initially believed.

There is a pronounced increase in the spread of RC.

So the distribution has changed...but it has mostly changed with regards to the extreme upward tails of the distribution. IOW, it seems to me that the elites of the modern era are subsantially out performing (or out run creating) the elites of the 80's in relation to the means of their distributions...however the bulk of players (i.e. most of the players) are distributed relatively simillar from era to era.

But is that really a problem of comparison? Maybe the elites of the ought are just that much better than the elites of the eighties.

I guess I'd like to see Val's argument threshed out a little more so I can determine if I disagree with it. ;) D.GOOCH

Cardinals, Cardinals, Cardinals!

by GOOCH24 on Feb 11, 2008 7:23 PM EST up reply actions 0 recs

My guess is that this phenomea

is directly related to the high average guys in the 2000s being much greater homerun threats than they were in the 80s. But as soon as you normalize out the homeruns, the sluggers of the 2000s will look pretty pedestrian. (BTW, that is basically what I have learned playing with whatifsports sim match up and sim leagues.)

As far as who is better, I guess we can't really determine for sure since both guys are in the upper tail of their respective distribution. That was my whole point when I said 3 or 4 of Mattingly's best years are comparable to Pujols.

The St. Louis Cardinals- 11 time World Champions!

by Zubin on Feb 11, 2008 9:49 PM EST up reply actions 0 recs

Note...

if you look at the data cutpoints (quantiles) included in the original post I think my observation bears out.

At the 100% cut point (i.e. the maximum RC for that decade) there is a significant difference between the 80's and the 00's:

2000's: 230
1980's: 153

That's a fairly substantial difference. The top score is 5.01 standard deviations above the mean, while the top for the 1980's is 3.63 SD's above the mean.

2000's: 5.01 SD's a/b mean
1980's: 3.63 SD's a/b mean

But that difference rapidly dissapates as we move past the very extreme of the distribution.

2000's: 1.89 SD's a/b mean
1980's: 1.93 SD's a/b mean

Note how the difference here has actually flipped. The 95 percentile in the 1980's actually outperformed the 2000's relative to their mean, though these are virtually identical.

Let's look at the 75th percentile:

2000's: 0.60 SD's a/b mean
1980's: 0.66 SD's a/b mean

Again, same result. 1980's slightly outperform the oughts, though difference is statistically irrelevant.

And the 25th percentile:

2000's: 0.82 SD's b/l mean
1980's: 0.83 SD's b/l mean

So I would argue that other than the absolute extreme of the 2000's distribution, the relative shape of the two distributions is substantially the same...presenting little to no problem for OPS+.

Additionally, I'm not convinced the extreme values are problematic for OPS+ either, since this is calculated relative to the mean. IOW, OPS+ should reflect the fact that the elite of the elite players in the oughts have been that much better than the average player in the oughts when compared to their counterparts in the 1980's.

Val? D.GOOCH

Cardinals, Cardinals, Cardinals!

by GOOCH24 on Feb 12, 2008 1:28 PM EST up reply actions 0 recs

I never really wanted to decry OPS+

it's a very useful tool. My problem, and I think these data establish it, is indicating the exceptionality of a player performing at a certain level. For me, the presence of Sammy Sosa makes the McGwire phenomenon less astounding than Babe Ruth's seasons, where the only competition was Rogers Hornsby hitting twenty less homers a season in the other league.

And I think it's valid concern for HOF consideration--not just ask the question "How much did the guy beat the average player", which OPS+ answers pretty well, but also ask "How many other guys performed at the same level?" or "How rare is it for a layer to perform at this level?"

And I think you've shown, at the extremes, these are empirically shown to not the exact same question.

by Valatan on Feb 12, 2008 1:37 PM EST up reply actions 0 recs

Disagree...

"So I would argue that other than the absolute extreme of the 2000's distribution, the relative shape of the two distributions is substantially the same...presenting little to no problem for OPS+."

I agree that your data validates using OPS+ for the middle ~95% or so of the distribution, but when we are discussing HoFers we are generally discussing guys in the upper 1-2%. And therefore OPS+ isn't as great a tool when measuring aross eras.

The St. Louis Cardinals- 11 time World Champions!

by Zubin on Feb 12, 2008 2:29 PM EST up reply actions 0 recs

But...

Here's my question, Zubin, don't we want our comparisons across eras, for HOF considerations, to be contexualized to the era in which the player played in?

As I see it, OPS+ is useful (whereas OPS may be much less so) because it factors in the mean OPS for that particular season (hence, contingent on the era of baseball the offense is produced in)...and thus accounts for the general inflation of offensive statistics in the 00's...as much as it accounts for the deflation of offensive statistics in the 70's and 80's.

I can see the problem with using OPS on its own. An .800 OPS in 2007 is much different than an .800 OPS in 1987. But OPS+ adjusts for the era effects by accounting for the average OPS in the league for that season.

So, as I see it, OPS+ should be an era-independent stat. D.GOOCH

Cardinals, Cardinals, Cardinals!

by GOOCH24 on Feb 12, 2008 3:46 PM EST up reply actions 0 recs

I think you misunderstand me

Obviously OPS+ is a better tool than OPS, but your analysis shows comparisons across eras break down once you hit the tails of the distribution.

Again for 95%+ of players it will work great, but when we are writting about Pujols and Mattingly it is not as meaningful of a comparison.

The St. Louis Cardinals- 11 time World Champions!

by Zubin on Feb 13, 2008 10:33 AM EST up reply actions 0 recs

Not to mention

with that rightward skew, there is some issue with commparison of players above the median to players below the median.

by Valatan on Feb 13, 2008 12:02 PM EST up reply actions 0 recs

Now that...

I think does present a problem for w/n era comparisons. A skewed distribution that is doing funny things with your mean can make a mean-dependent stat biased.

If we're interested in looking at offensive production relative to the middle offensive player (in a season, era, what have you) rather than the average offensive production, then we might recast OPS+ using the league median rather than the league mean OPS. The median, of course, is immune to tail effects. Call it OPS++. ;) D.GOOCH

Cardinals, Cardinals, Cardinals!

by GOOCH24 on Feb 13, 2008 5:30 PM EST up reply actions 0 recs

OK...

...I understand that the figures I've provided show that players in the 00's outperformed players in the 80's relative to the mean RC player.

My question is: why is that a problem of comparison? If OPS+ shows that Pujols was, say, 30% better than Mattingly...then it would tend to conform with the distributional differences we've noted between the eras.

So I guess my questions are this:

Do you think OPS+ would show that Mattingly is worse than Pujols?

If that's what OPS+ shows, why would that be a problem for era comparisons?

If the problem is the mean being drawn up by the tails, I would expect the opposite problem: OPS+ would show the difference between Pujols & Mattingly to be smaller than it should be.

Is that what you're arguing?

D.GOOCH

Cardinals, Cardinals, Cardinals!

by GOOCH24 on Feb 13, 2008 5:35 PM EST up reply actions 0 recs

MHO...

I think it is a problem of comparison because there are at least two ways to look at this. The first is to compare each to the mean, the second is to compare them on a percentile basis. The first may tell you how good each is, but the second tells you how exceptional each is.

As far as comparisons, I would expect OPS+ to show in their best seasons, Pujols was better than Mattingly.

That being said, OPS+ still presents a problem for comparisons because the environment is different and the populations you are basing your comparison upon are different.

I am not so sure about the means drawing up the tails. I think the problem are the tails themselves. Again once we hit the area where the distribution are no longer comparable, I think we lose the ability to compare players across eras.

The St. Louis Cardinals- 11 time World Champions!

by Zubin on Feb 14, 2008 3:17 PM EST up reply actions 0 recs

Also,

the deviation going from 28 to 34 from the 80s to the 00s isn't really that small of a change--that's 20%. That's the difference between an OPS+ of 150 and an OPS+ of 182, if we're just casually multiplying. The actual effect will be smaller, probably significantly so, but still, variations in the sd on the order of 10-20% can really affect the ends of a distribution.

by Valatan on Feb 11, 2008 2:49 PM EST up reply actions 0 recs

And additionally,

José Valentin is commenting at VeB? Hello to our first MLBer! ;)

by Valatan on Feb 11, 2008 2:52 PM EST up reply actions 0 recs

whoops...

Sorry there, Valatan. Must have had Jose Valentin on the brain. ;) D.GOOCH

Cardinals, Cardinals, Cardinals!

by GOOCH24 on Feb 11, 2008 3:31 PM EST up reply actions 0 recs

Now for gits and shiggles

I think it would be interesting to look at this year by year on a graph, with the years of expansion and realignment notated.

Would also be interesting to see that graph overlayed with the average ERA...

Excellent work Gooch. I take back all the average things I've said about you. :)

"Dude, we're running out of stadium" - said on the way to our seats in Section 428.

by bukowski on Feb 10, 2008 11:06 PM EST reply actions 0 recs

Excellent Post

This is the reason I read this site everyday. Great analysis and food for thought.

by ajo080s on Feb 11, 2008 8:37 AM EST reply actions 0 recs

Just to be clear

you are looking at the player average in Runs Created, not the league average. I don't know which average OPS+ uses. It may be a slight difference, but its a difference (depending on how many players have AB<150).

Also, if the distribution suffers from censoring, just use a censored normal distribution estimate of the standard deviation and mean.

http://www.ssicentral.com/lisrel/techdocs/censor.pdf

There's no easy function to do this in Excel, if that's what you're using, but it would only take a couple more cells to fill in.

by enoscountry on Feb 11, 2008 3:28 PM EST reply actions 0 recs

some answers...

The RC calculation is the average of the player's RC.

You're right, if censoring is the problem there is a correction factor that can be applied to get unbiased SD's and means.

I'm using SAS. D.GOOCH

Cardinals, Cardinals, Cardinals!

by GOOCH24 on Feb 11, 2008 7:26 PM EST up reply actions 0 recs

Interesting

My knee jerk thought is that RC is OPS. If that is unadjusted, then I'm not sure what we are seeing here. I seriously just glanced down the page, but you are essentially looking at which era has a higher offense, right? The "+" part of OPS+ is already describing the run environment.

We're just seeing how many elite hitters and vice versa were around in each decade. You could just do this with a graph of OPS+ by decade to see how heavy the tails are or whatever.

Also, the decade cutoffs are arbitrary. 1967 to 1977 might get you a totally different answer because it causes two elite or terribly shitty player's peak or valley seasons to overlap when they did not.

by plh903 on Feb 13, 2008 2:47 AM EST reply actions 0 recs