Standard deviation of team run differential

[brett]

Has anyone compiled standard deviations among team run differentials for MLB seasons?

To be clear, let's say there are 8 teams in the league. For each team, take runs scored minus runs allowed. This is the variable that I would like to see the league standard deviation for.

I think I read that people have looked at it as a sign of league quality, so I was hoping it has been done. If not, someone might like to take it as a project. Doesn't seem too extensive.

Also, I saw a stat on BBRef: defensive efficiency. Is that just 1-BABIP?

[Tyrus4189Cobb]

To be clear, let's say there are 8 teams in the league. For each team, take runs scored minus runs allowed. This is the variable that I would like to see the league standard deviation for.

I'm not sure what this means. Subtracting RA from RS wouldn't be a measure of deviation. It's just a difference. Do you mean find the deviation based on those numbers?

As I learned it, standard deviation is the units (measured in std devs) for which a figure deviates from the average. http://www.socialresearchmethods.net...s/statdes3.gif

We would find the run differentials for each team, average them (will always be 0), then subtract the average from each sample (using absolute value because we are only measuring distance). Add all of these up, divide by the entire sample set minus one, and finish by taking the square root. Is this what you're looking for?

1901 AL:
Orioles score 761, allow 750 (+11)
Americans score 759, allow 608 (+151)
White Sox score 819, allow 632 (+187)
Blues score 666, allow 831 (-165)
Tigers score 742, allow 696 (+46)
Brewers score 641, allow 828 (-187)
Athletics score 805, allow 760 (+45)
Senators score 683, allow 771 (-88)

Average run differential: 0

(11+151+187+165+46+187+45+88+0)=880
divided by sample size minus one=110.
Square root= about 10.5. Standard deviation is 10.5 runs.

Is this what you mean?

[Bothrops Atrox]

Also, I saw a stat on BBRef: defensive efficiency. Is that just 1-BABIP?

Yes. Not sure what they do with bunt singles or sacrifice flies, however.

[brett]

I'm not sure what this means. Subtracting RA from RS wouldn't be a measure of deviation. It's just a difference. Do you mean find the deviation based on those numbers?

As I learned it, standard deviation is the units (measured in std devs) for which a figure deviates from the average. http://www.socialresearchmethods.net...s/statdes3.gif

We would find the run differentials for each team, average them (will always be 0), then subtract the average from each sample (using absolute value because we are only measuring distance). Add all of these up, divide by the entire sample set minus one, and finish by taking the square root. Is this what you're looking for?

1901 AL:
Orioles score 761, allow 750 (+11)
Americans score 759, allow 608 (+151)
White Sox score 819, allow 632 (+187)
Blues score 666, allow 831 (-165)
Tigers score 742, allow 696 (+46)
Brewers score 641, allow 828 (-187)
Athletics score 805, allow 760 (+45)
Senators score 683, allow 771 (-88)

Average run differential: 0

(11+151+187+165+46+187+45+88+0)=880
divided by sample size minus one=110.
Square root= about 10.5. Standard deviation is 10.5 runs.

Is this what you mean?

that is the set of data that I want, but you square each team's run differential, add them together, divide by the number of teams and then take the square root like this:

1901 AL:
Orioles score 761, allow 750 (+11) squared=+121
Americans score 759, allow 608 (+151) squared=+22801
White Sox score 819, allow 632 (+187) squared=+34969
Blues score 666, allow 831 (-165) squared=+27225 (all squares are positive)
Tigers score 742, allow 696 (+46) squared=: +2116
Brewers score 641, allow 828 (-187) squared=: +34969
Athletics score 805, allow 760 (+45) squared=: +2025
Senators score 683, allow 771 (-88) squared=: +7744
sum of all squared values= 131,970
Divided by 8=16496.25
Square root=128.4 so the standard deviation would be 128.4

[Tyrus4189Cobb]

Yup, 128.4 is correct. I forgot to square the differentials. How does it relate to league quality?

[Jackaroo Dave]

League QUALITY? Well, in a completely open league, where anyone of any age could play, you would probably have a greater disparity between teams than in a highly filtered league with a high threshold. But there are lots of other factors at work. For example, high minor leagues might have a lower standard deviation of run differentials than majors because of a negative feedback effect: Once a player is established as outstanding, he leaves the league. And the worst players are replaced from below.

In general, wins and losses line up pretty well with run differential, so it seems like you are asking whether balanced or unbalanced leagues are of higher quality. What is the lure of using run differential instead of just won-lost records?

(edit) Sorry, Ty, I see you were asking brett, who would actually know the answer, not throwing it out for the group. Sorry, brett, for butting in.

[brett]

Maybe it has nothing to do with LQ. I'm interested anyway. What the variation is large though because a few teams have a large share of the best players? It would suggest to me that that would tend to trap more value on their benches and in their minors, while the bottom teams might be using replacement level guys. If you balance things out, there are fewer MLB quality guys on benches or in the minors of good teams.
Plus if we are talking about the highest league, (versus minors) you aren't going to lose good players to a better league.

Even if it has no direct connection, I think it would be interesting to see anyway.

Anyone want to do it just for one year per decade to see if there are trends?

[Tyrus4189Cobb]

Anyone want to do it just for one year per decade to see if there are trends?

I'll do it. Gimme a bit

[Tyrus4189Cobb]

1905	RS	RA	Diff	Dff^2
BOS	578	564	14	196
CHW	614	450	164	26896
CLE	563	587	-24	576
DET	511	608	-97	9409
NYH	588	622	-34	1156
PHA*	623	492	131	17161
SLB	509	608	-99	9801
WSH	560	623	-63	3969
				8645.5
Std Dev				92.98
				
1915	RS	RA	Diff	Dff^2
BOS*	668	499	169	28561
CHW	717	509	208	43264
CLE	539	669	-130	16900
DET	778	597	181	32761
NYY	583	588	-5	25
PHA	545	889	-344	118336
SLB	521	679	-158	24964
WSH	570	491	79	6241
				33881.5
STD Dev				184.07
				
1925	RS	RA	Diff	Dff^2
BOS	640	921	-281	78961
CHW	811	773	38	1444
CLE	783	817	-34	1156
DET	901	828	73	5329
NYY	706	774	-68	4624
PHA	831	713	118	13924
SLB	898	907	-9	81
WSH*	828	670	158	24964
				16310.375
STd Dev				127.71
				
1935	RS	RA	Diff	Dff^2
BOS	718	732	-14	196
CHW	738	750	-12	144
CLE	776	739	37	1369
DET*	919	665	254	64516
NYY	818	632	186	34596
PHA	710	869	-159	25281
SLB	718	930	-212	44944
WSH	823	902	-79	6241
				22160.875
STD Dev				148.87
				
1945	RS	RA	Diff	Dff^2
BOS	599	674	-75	5625
CHW	596	633	-37	1369
CLE	557	548	9	81
DET*	633	565	68	4624
NYY	676	606	70	4900
PHA	494	638	-144	20736
SLB	597	548	49	2401
WSH	622	562	60	3600
				5417
Std Dev				73.60
				
1955	RS	RA	Diff	Dff^2
BAL	540	754	-214	45796
BOS	755	652	103	10609
CHW	725	557	168	28224
CLE	698	601	97	9409
DET	775	658	117	13689
KCA	638	911	-273	74529
NYY*	761	569	192	36864
WSH	598	789	-191	36481
				31950.125
STD Dev				178.75
				
1965	RS	RA	Diff	Dff^2
BAL	641	578	63	3969
BOS	669	791	-122	14884
CAL	527	569	-42	1764
CHW	647	555	92	8464
CLE	663	613	50	2500
DET	680	602	78	6084
KCA	585	755	-170	28900
MIN*	774	600	174	30276
NYY	611	604	7	49
WSA	591	721	-130	16900
				11379
STd Dev				106.67
				
1975	RS	RA	Diff	Dff^2
BAL	682	553	129	16641
BOS*	796	709	87	7569
CAL	628	723	-95	9025
CHW	655	703	-48	2304
CLE	688	703	-15	225
DET	570	786	-216	46656
KCR	710	649	61	3721
MIL	675	792	-117	13689
MIN	724	736	-12	144
NYY	681	588	93	8649
OAK**	758	606	152	23104
TEX	714	733	-19	361
				11007.333
STD Dev				104.92
				
1985	RS	RA	Diff	Dff^2
BAL	818	764	54	2916
BOS	800	720	80	6400
CAL	732	703	29	841
CHW	736	720	16	256
CLE	729	861	-132	17424
DET	729	688	41	1681
KCR*	687	639	48	2304
MIL	690	802	-112	12544
MIN	705	782	-77	5929
NYY	839	660	179	32041
OAK	757	787	-30	900
SEA	719	818	-99	9801
TEX	617	785	-168	28224
TOR**	759	588	171	29241
				10750.143
STD Dev				103.68
				
1996	RS	RA	Diff	Dff^2
BAL***	949	903	46	2116
BOS	928	921	7	49
CAL	762	943	-181	32761
CHW	898	794	104	10816
CLE**	952	769	183	33489
DET	783	1103	-320	102400
KCR	746	786	-40	1600
MIL	894	899	-5	25
MIN	877	900	-23	529
NYY*	871	787	84	7056
OAK	861	900	-39	1521
SEA	993	895	98	9604
TEX**	928	799	129	16641
TOR	766	809	-43	1849
				15746.857
STD Dev				125.47
				
2005	RS	RA	Diff	Dff^2
BAL	729	800	-71	5041
BOS***	910	805	105	11025
CHW*	741	645	96	9216
CLE	790	642	148	21904
DET	723	787	-64	4096
KCR	701	935	-234	54756
LAA**	761	643	118	13924
MIN	688	662	26	676
NYY**	886	789	97	9409
OAK	772	658	114	12996
SEA	699	751	-52	2704
TBD	750	936	-186	34596
TEX	865	858	7	49
TOR	775	705	70	4900
				13235.143
STd Dev				115.04

Where * denotes pennant winner, ** denotes division winner, *** denotes wildcard. Ideally, lgRS+lgRA=0. Any discrepencies are due to missing RetroSheet data or IL play. If you need any more, let me know. I made a program to calculate it with relative ease.

[Jackaroo Dave]

Maybe it has nothing to do with LQ. I'm interested anyway. What the variation is large though because a few teams have a large share of the best players? It would suggest to me that that would tend to trap more value on their benches and in their minors, while the bottom teams might be using replacement level guys. If you balance things out, there are fewer MLB quality guys on benches or in the minors of good teams.
Plus if we are talking about the highest league, (versus minors) you aren't going to lose good players to a better league.

Oh, yeah, I suggested the high minors as an exception to the general rule that LQ varies inversely to competitive balance: The quality of play is poorer than the big leagues, but the run diff SD is, maybe, smaller.

In his article on why there are no .400 hitters, Stephen Gould suggests that it's because inter-player variation has shrunk, so a smaller BA gap now represents a larger SD difference. And this in turn is because baseball has evolved. Just like cars: there's much less variation in auto body shape, because they are all approaching the same aerodynamic ideal.

[Jackaroo Dave]

Run differential might track runs scored. I know it doesn't linearally--I was surprised how little difference in run diff there was between high- and low-scoring eras. But there is probably some effect. I would guess the absolute diff would increase for both the best and the worst teams.

Another possibly troublesome factor is that changes like expansion, contraction, integration, and war affect both quality and standard deviation: Expansion reduces overall quality by bringing in lower-level players, and by the same token increases individual variation. If the distribution of new players is not equitable (as in the 60s), then variation at the team level would increase. League contraction works in the opposite directions, and again, I would guess that some teams can take more advantage of this than others. (Ditto integration. War is tougher to figure out.)

So you have quality and variation moving in opposite directions, but neither one an independent variable, both dependent on big, sudden changes in league structure.

Not to horn in on this, but Ty, if it really is no trouble, why not run every year? For example, if we had 1913-16, we could look for any effects due to the Federal League.

[Bothrops Atrox]

1905	RS	RA	Diff	Dff^2
BOS	578	564	14	196
CHW	614	450	164	26896
CLE	563	587	-24	576
DET	511	608	-97	9409
NYH	588	622	-34	1156
PHA*	623	492	131	17161
SLB	509	608	-99	9801
WSH	560	623	-63	3969
				8645.5
Std Dev				92.98
				
1915	RS	RA	Diff	Dff^2
BOS*	668	499	169	28561
CHW	717	509	208	43264
CLE	539	669	-130	16900
DET	778	597	181	32761
NYY	583	588	-5	25
PHA	545	889	-344	118336
SLB	521	679	-158	24964
WSH	570	491	79	6241
				33881.5
STD Dev				184.07
				
1925	RS	RA	Diff	Dff^2
BOS	640	921	-281	78961
CHW	811	773	38	1444
CLE	783	817	-34	1156
DET	901	828	73	5329
NYY	706	774	-68	4624
PHA	831	713	118	13924
SLB	898	907	-9	81
WSH*	828	670	158	24964
				16310.375
STd Dev				127.71
				
1935	RS	RA	Diff	Dff^2
BOS	718	732	-14	196
CHW	738	750	-12	144
CLE	776	739	37	1369
DET*	919	665	254	64516
NYY	818	632	186	34596
PHA	710	869	-159	25281
SLB	718	930	-212	44944
WSH	823	902	-79	6241
				22160.875
STD Dev				148.87
				
1945	RS	RA	Diff	Dff^2
BOS	599	674	-75	5625
CHW	596	633	-37	1369
CLE	557	548	9	81
DET*	633	565	68	4624
NYY	676	606	70	4900
PHA	494	638	-144	20736
SLB	597	548	49	2401
WSH	622	562	60	3600
				5417
Std Dev				73.60
				
1955	RS	RA	Diff	Dff^2
BAL	540	754	-214	45796
BOS	755	652	103	10609
CHW	725	557	168	28224
CLE	698	601	97	9409
DET	775	658	117	13689
KCA	638	911	-273	74529
NYY*	761	569	192	36864
WSH	598	789	-191	36481
				31950.125
STD Dev				178.75
				
1965	RS	RA	Diff	Dff^2
BAL	641	578	63	3969
BOS	669	791	-122	14884
CAL	527	569	-42	1764
CHW	647	555	92	8464
CLE	663	613	50	2500
DET	680	602	78	6084
KCA	585	755	-170	28900
MIN*	774	600	174	30276
NYY	611	604	7	49
WSA	591	721	-130	16900
				11379
STd Dev				106.67
				
1975	RS	RA	Diff	Dff^2
BAL	682	553	129	16641
BOS*	796	709	87	7569
CAL	628	723	-95	9025
CHW	655	703	-48	2304
CLE	688	703	-15	225
DET	570	786	-216	46656
KCR	710	649	61	3721
MIL	675	792	-117	13689
MIN	724	736	-12	144
NYY	681	588	93	8649
OAK**	758	606	152	23104
TEX	714	733	-19	361
				11007.333
STD Dev				104.92
				
1985	RS	RA	Diff	Dff^2
BAL	818	764	54	2916
BOS	800	720	80	6400
CAL	732	703	29	841
CHW	736	720	16	256
CLE	729	861	-132	17424
DET	729	688	41	1681
KCR*	687	639	48	2304
MIL	690	802	-112	12544
MIN	705	782	-77	5929
NYY	839	660	179	32041
OAK	757	787	-30	900
SEA	719	818	-99	9801
TEX	617	785	-168	28224
TOR**	759	588	171	29241
				10750.143
STD Dev				103.68
				
1996	RS	RA	Diff	Dff^2
BAL***	949	903	46	2116
BOS	928	921	7	49
CAL	762	943	-181	32761
CHW	898	794	104	10816
CLE**	952	769	183	33489
DET	783	1103	-320	102400
KCR	746	786	-40	1600
MIL	894	899	-5	25
MIN	877	900	-23	529
NYY*	871	787	84	7056
OAK	861	900	-39	1521
SEA	993	895	98	9604
TEX**	928	799	129	16641
TOR	766	809	-43	1849
				15746.857
STD Dev				125.47
				
2005	RS	RA	Diff	Dff^2
BAL	729	800	-71	5041
BOS***	910	805	105	11025
CHW*	741	645	96	9216
CLE	790	642	148	21904
DET	723	787	-64	4096
KCR	701	935	-234	54756
LAA**	761	643	118	13924
MIN	688	662	26	676
NYY**	886	789	97	9409
OAK	772	658	114	12996
SEA	699	751	-52	2704
TBD	750	936	-186	34596
TEX	865	858	7	49
TOR	775	705	70	4900
				13235.143
STd Dev				115.04

.

Random fluctuation or does somebody see anything obvious that correlated to the 1915, 1945, 1955 season outliers?

[Jackaroo Dave]

A couple of other issues or points of interest: It probably doesn't matter unless one tries to work out a specific relationship between the variations in league quality and run differentials, but . . . .

Run differential values aren't independent. Obviously, a plus run for one team is a minus run for another. So you wind up with twice as many differential runs as there are actual runs, just as you do with wins and losses. I think counting a run scored as half a differential run, and a run allowed as minus half, just as a win counts as a half game, might make sense.

I'm a little antsy about measuring variation with the standard deviation for the following reasons: The standard deviation is nice because the standard deviation of the sample mean typically has a normal distribution, whose own standard deviation, aka standard error, aka root mean error, is equal to the sample standard deviation divided by the square root of the sample size. That helps you make infernces and estimates about the population mean. As a purely descriptive statistic, however, it is no better than the mean absolute deviation (the average of the absolute value of the each team's difference from the mean).

But in this case, the sample mean is not a variable: it's a constant; it's always equal to the population mean, zero. The number of runs ahead is equal to the number of runs behind. The sample mean has no standard deviation, because it's a constant. Consequently, no kind of statistical work connecting Run Differential to another variable can be "parametric," i.e. based on the assumption that the sample mean is a normally distributed random variable with a certain mean and a certain standard deviation.

[Tyrus4189Cobb]

Not to horn in on this, but Ty, if it really is no trouble, why not run every year? For example, if we had 1913-16, we could look for any effects due to the Federal League.

That will take some time because I have to input every year. I can't do it right now but I'll get on it ASAP.

[brett]

Random fluctuation or does somebody see anything obvious that correlated to the 1915, 1945, 1955 season outliers?

well, oddly I tohught '45 would be high because of the war. Pull out x percent and bring up the next guy. Plus didn't they deaden the ball?