On our main project here. . . on Friday and part of Saturday I was working an idea to perhaps re-do some of what we have done and get better results.   We base the "floor"—the zero-competence line—on the number that is four standard deviations below the norm FOR THE DECADE; for teams in the 1950s, we use the 1950s to establish the zero-competence line.  

I thought perhaps I might get better results by doing something else, specifically this.  Suppose that, rather than placing each team into one of twelve decades, I placed each team into one of 24 "peer groups".   First I divided baseball history into six twenty-year periods, 1900-1919, 1920-1939, 1940-1959, 1960-1979, 1980-1999, or 2000-2019.   Each of those six groups I divided into four "run levels", based on the runs scored/game in the league—high scoring, above average scoring, below average scoring level, low scoring.  In that way, all 241 leagues were placed into groups of leagues from (a) the same period, and (b) with a reasonably similar runs scored level. 

I thought that that might make the zero-competence level more predictable, more standard, thus stabilize the data, thus reduce the prediction error at the outcome.

I worked on it probably 15-20 hours, but it just didn’t work.  At all.   The prediction error was not only larger, it was MUCH larger; twice as large.  Some of that I could have gotten rid of by re-doing OTHER parts of the system, no doubt, but it was clear that I wasn’t going to gain anything without re-doing hundreds of hours of work, so I gave it up.  

Happens all the time.  It’s just research; anybody who does research, any field, knows that that’s the way it goes.  You invest an immense percentage of your work time in approaches that ultimately don’t lead anywhere.   It’s common.  The only difference here is that I am doing in public here things that are usually done quietly, without public documentation, so I have to report that I don’t have anything to report.   Wasted days.

 

 

 

The Learning Curve Simulator

            Then I took the rest of the weekend off to work on a couple of little fun projects.   This is based on an idea that I had decades ago, but have never written about before, I don’t think.   I had played around with it before, but for some reason this time something clicked, and I got a little bit out of it. 

            I created what I called the Learning Curve Simulator.  Suppose that you have a player who enters major league baseball with a "known" set of skills, known to us, although presumably unknown to his team, his fans, and even the athlete himself.   The athlete in any given plate appearance has a given probability of hitting a single, a given probability of hitting a double, a triple, a home run, etc.  Seven outcomes:  Walk, Single, Double, Triple, Home Run, Strikeout, Undescribed Out.   The player is defined by those seven outcomes, and those seven outcomes occur at random. 

            The thing is, though, that as the player goes through plate appearances, his skills grow and change according to his previous at-random outcomes.  If a player happens to hit a home run, then that home run becomes a part of his established skill set, and the probability that he will hit a home run increases by some very small amount.  It is unclear whether we are modeling confidence of learning here; they’re indistinguishable in practice; you can’t develop confidence in your ability to do something unless you know how to do it, and you can’t learn how to do it well and predictably until you develop some confidence.   After a player hits a home run, he gains a little bit of confidence in his ability to hit home runs, and also he gains a little bit of knowledge about how to hit a home run; it becomes a part of his skill set.  When he strikes out, he loses confidence, and becomes more likely to strike out in the next plate appearance, and in all future plate appearances. 

            The player’s "skills" or his "starting set skills" are the same from one iteration of the study to the next.   The player’s career is different every time you hit the "go" button, however, because the random numbers in the system trigger random events, and each random event becomes a predictor for subsequent events.  His career is, in a sense, a Random Walk, and I think that I have been working out how to do this since I first heard of the concept of a Random Walk, which was sometime in the 1980s.   But the computers we had then did not have the space, the memory, to construct an experiment like this.   Probably didn’t have enough space to do it until maybe five years ago, and didn’t get around to it until now. 

            Anyway, the "starting point" for each iteration was a player who, per 1,000 at bats, would get

            70 walks

            230 hits (a .247 batting average)

            162 singles  

            40 Doubles

            10 Triples

            18 Homers

            200 strikeouts, and

            500 undescribed outs. 

            Which gives him a .300 on base percentage, .370 slugging percentage, .670 OPS.   In other words, this is a marginal major leaguer, can probably have a career if he is a defensive wizard and figures it out a little at the plate.

            This player, then, is exposed a set of random numbers.  The first random number determines whether he gets the plate appearance or does not.   The formula for this is his OPS (starts out at .670), minus .550, divided by 200.  That works out to .600, so for his first plate appearance, there is a 60% probability that he will get the plate appearance, a 40% chance that he won’t.   He is not guaranteed playing time, in other words.  He gets playing time if he DESERVES playing time, doesn’t get it if he doesn’t deserve it.

            After the first plate appearance, and with each plate appearance, his OPS either goes up, or it goes down.   If his OPS goes up, his playing time increases; if his OPS goes down, his playing time decreases. 

            Then, if the player gets the at bat, another random number generates an outcome from the at bat.   That outcome then replaces one one-thousandth of his known skill set, creating a newly defined skill set. 

            The question which we are ultimately pursuing here, although we are miles away from any answers,  is "To what extent are a player’s skills malleable or plastic, and to what extent are they limited or determinate?"   Well. . .we know that player’s skills are infinitely malleable on the negative slope, because there is this thing called "death"; if you die, you lose all of your ability.  There are lots of things that can happen to you to make you a LESS GOOD player.  What we don’t really understand is to what extent you can get better. 

            So, in my first attempt to make this work, I assumed that the player’s "present skill level" was based on:

1)      His last 1,000 plate appearances, plus

2)     His last 500 plate appearances, plus

3)     His last 100 plate appearances, plus

4)     His last 10 plate appearances, and

5)     He comes to the majors with a "silent history" of 1,000 plate appearances, presumably in the minor leagues, which are now represented as identical to one another, the player getting .23 hits, .04 doubles, .01 triples, .018 homers, .07 walks, etc., in each of those plate appearances. 

In other words, when a player is new to the major leagues, he is not "new"; he has A history.   It is just not a major league history.

What we are assuming here is that what has happened to the player in the last year (500 PA) is more relevant to where he is now than what happened the previous year, and also that what has happened in the last month (100 PA) carries a little more relevance, and also that what is fresh in the player’s memory (10 PA) carries even more relevance. 

The player’s skills, then, are represented in 1,610 plate appearances, which are his last 1,000 plate appearances, but with the last 500 counted at 2, the last 100 counted at 3, and the last 10 counted at 4.  1000 + 500 + 100 + 10 = 1610.  In the player’s first major league plate appearance, his chance of hitting a home run is 18/1000, or 28.98/1610.   If he happens to hit a home run in his first PA, this increases to 32.098/1610, or 20.44/1000—but the increase is temporary.  If he doesn’t hit another home run in his next nine at bats, which he probably will not, then his production level starts to slide back to where it was. 

I don’t know to what extent you are following me here, but I’m doing the best I can to explain.   Anyway, simulating plate appearances with that set of assumptions, I generated 13-year "careers" for each player, with the last season being just a little more than a half-season.   Each 720 potential plate appearances was a season, remembering that the player doesn’t get ALL of those 720 plate appearances unless he drives his OPS over .750.   And if the player’s OPS for his last 1000 plate appearances drops below .550, then he drops out of the majors.

I was interested to see:  will some of these players, with this set of assumptions, become superstars, and post .900 career OPS?  Or will that not happen. 

What happened in the first set of test runs is that the careers quickly became unrealistic.  One thing that would happen a lot is that players would start to hit unrealistic numbers of triples—35, 40 triples in a season sometimes.  And there were other unrealistic patterns, players hitting 70 doubles in a season, but the triples one was the most notable because the expected number of triples is so low that that is the easiest one to wander outside of it’s normal bounds. 

So my first thought was, "Well, the player doesn’t really "learn" to hit triples in the same way that he learns to hit home runs or learns to take a walk.  The triple is just something that happens sometimes; it is not a predictable outcome of a given swing path."  So my first effort to fix that problem was to divert half of the "learned triples" into doubles; in other words, when a player happened to hit a triple, that increased his chance of hitting a double or a triple by a little bit, but had less impact on the triple itself.

That didn’t really do any good; that just caused some players to hit unrealistic numbers of doubles AND triples, and made those effects a little bit more muted so that the OTHER anomalies in the system would become more noticeable.   What I learned from this is:  player’s skills are not THAT malleable.  My initial set of assumptions had overstated the extent to which a player might reasonably learn from his major league experience. 

I started adding "shaping" to the player’s history; in other words, I started adding into the 1,610-plate-appearance image of the player’s current and evolving skills a small number of plate appearances that don’t change no matter what the player has done.   The game has a normal shape that no player can escape.  Adding to each plate appearance a little bit of "normal shape" reduced the likelihood that the player’s career would wander outside of those normal parameters. 

The shaping was based on the player’s starting point—that is, per 1,000 plate appearances, he would have 230 hits, 40 doubles, 10 triples, etc.   I first put into the system 100 plate appearances of "normal shaping", thus increasing the base sample for each plate appearance to 1,710 plate appearances—1,000, plus 500, plus 100, plus 10, plus 100 plate appearances of immutable shaping. 

This proved to be inadequate.  Player’s careers still wandered outside of normal parameters.  I increased the shaping to 200 PA; still not enough.  I increased it to 400 PA; still not enough.   I increase it to 600 PA; that seemed to work.   That was enough to keep the player’s skills within normal ranges.

So now the player’s skill set was defined by 2,210 plate appearances—1,000, plus 500, plus 100, plus 10, plus 600.    That is as far as the experiment went, and all of the "imaginary careers" that I will show you at the end of this article were generated using this hypothesis.  This premise, this approach; whatever you want to call it. 

These studies, primitive as they were, produced reasonably realistic-looking careers, not truly realistic, and I did learn something from doing them.   I got a sort of "first look" at the extent to which a player can adapt and re-shape his talents on the upside, and some understanding of the role.

At this point, our failures and limitations are more significant than our successes—which, again, is common in research; the most common thing you learn from doing research is why your research failed.  Anyway, the most obvious false assumption of this research is that the player’s skills remain "plastic" or "adaptable" throughout his career at the same level they are when he enters the majors.   This causes the "projected careers" of these players to very often take unprecedented and frankly improbable twists six, eight, ten years into the career.  Those same changes-of-direction might not be improbable early in the players career; they are later.  My model doesn’t build that in. 

I ran several hundred "test career projections" with this model, and captured a few of them to share with you.   The display models are not fully representative; I tented to save the ones in which the player did better.  There are a lot of very short and unsuccessful career, and there are a lot of tests in which the "player" has the full 13-year career but winds up with a career OPS in the mid-.600s.  There are not many "good" careers that develop out of this premise, and there are no great ones.  One player, whose career I failed to capture, hit 200-some home runs and wound up with a career OPS in the .770-.780 range.  But no player wound up with a career .300 average or 500 homers or anything remotely like that; I think just the one player with over 200.   The model suggests, although it certainly does not prove, that the extent to which a player may improve his skills at the major league level is limited. 

Of course, in real life, the "starting point" is a variable, not a constant.   I was attempting here to look at the question of variability FROM A CONSTANT STARTING POINT.  Among real players, the starting point is not a constant. 

I also have another little study ready to write up here, but it is getting late enough in the day that I’m not sure I will have it ready by 5 o’clock.  I’ll go ahead and post this one, and then work on the other one.   Thanks for reading.

 

AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
278	52	10	1	3	13	71	.187	.223	.263	.486
138	32	5	0	0	7	42	.232	.269	.268	.537
24	4	1	0	0	1	9	.167	.200	.208	.408
440	88	16	1	3	21	122	.200	.236	.261	.498
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
495	133	20	1	17	38	104	.269	.321	.416	.737
330	72	9	0	7	23	65	.218	.269	.309	.578
407	104	9	5	17	34	77	.256	.313	.428	.740
403	101	16	5	5	21	83	.251	.288	.352	.640
503	140	28	4	19	31	83	.278	.320	.463	.783
482	112	13	1	10	31	83	.232	.279	.326	.604
349	94	22	3	6	34	75	.269	.334	.401	.735
317	65	12	4	2	26	70	.205	.265	.287	.552
298	85	14	0	6	14	54	.285	.317	.393	.710
444	118	30	8	2	34	98	.266	.318	.383	.701
377	79	18	7	4	18	82	.210	.246	.326	.572
193	37	5	3	3	22	46	.192	.274	.295	.570
200	50	7	9	3	11	52	.250	.289	.420	.709
4798	1190	203	50	101	337	972	.248	.297	.374	.672
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
385	98	7	4	11	36	78	.255	.318	.379	.698
351	81	9	2	5	10	70	.231	.252	.311	.563
181	35	5	2	1	11	37	.193	.240	.260	.499
245	58	14	1	5	23	45	.237	.302	.363	.666
456	121	34	2	7	33	110	.265	.315	.395	.710
438	116	23	1	8	16	98	.265	.291	.377	.667
431	120	24	3	4	16	93	.278	.304	.376	.680
417	118	25	5	3	19	78	.283	.314	.388	.703
260	55	13	1	3	11	65	.212	.244	.304	.547
428	107	21	1	22	21	100	.250	.285	.458	.743
401	90	11	0	10	34	98	.224	.285	.327	.612
525	163	31	2	16	38	115	.310	.357	.469	.826
326	92	27	2	5	15	81	.282	.314	.423	.737
4844	1254	244	26	100	283	1068	.259	.300	.382	.682
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
444	120	20	5	7	38	101	.270	.328	.385	.713
548	165	28	3	6	41	107	.301	.350	.396	.746
355	82	12	6	0	29	79	.231	.289	.299	.588
208	40	5	1	4	12	56	.192	.236	.284	.520
205	43	7	3	4	12	54	.210	.253	.332	.585
220	39	8	6	2	14	66	.177	.226	.295	.522
261	55	15	4	5	11	54	.211	.243	.356	.599
198	40	13	0	1	14	36	.202	.255	.283	.538
162	35	6	3	0	8	44	.216	.253	.290	.543
305	76	17	2	6	19	70	.249	.293	.377	.670
273	67	13	1	2	14	79	.245	.282	.322	.605
326	86	25	3	2	23	88	.264	.312	.377	.690
298	101	20	6	2	21	54	.339	.382	.466	.849
3803	949	189	43	41	256	888	.250	.297	.354	.651
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
413	95	13	8	15	27	85	.230	.277	.409	.686
383	93	16	5	4	38	89	.243	.311	.342	.653
329	84	19	1	4	25	61	.255	.308	.356	.664
424	109	27	1	5	21	93	.257	.292	.361	.653
157	20	6	0	2	11	42	.127	.185	.204	.388
318	90	27	4	6	40	65	.283	.363	.450	.813
635	178	41	9	6	40	151	.280	.323	.402	.725
412	115	26	3	1	33	93	.279	.333	.364	.697
316	73	13	1	1	25	61	.231	.287	.288	.575
253	65	11	0	0	25	57	.257	.324	.300	.624
251	55	10	1	2	18	60	.219	.271	.291	.562
285	73	15	3	0	17	71	.256	.298	.330	.628
154	43	12	0	0	9	28	.279	.319	.357	.676
4330	1093	236	36	46	329	956	.252	.305	.355	.661
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
431	115	29	3	5	27	84	.267	.310	.383	.693
256	55	7	1	2	14	51	.215	.256	.273	.529
222	47	7	0	1	18	49	.212	.271	.257	.528
144	31	5	3	1	16	28	.215	.294	.312	.606
506	147	25	6	13	34	104	.291	.335	.441	.776
570	166	37	6	5	38	109	.291	.336	.404	.739
395	95	21	1	2	39	58	.241	.309	.314	.623
408	103	19	1	9	50	67	.252	.334	.370	.704
493	131	20	1	23	44	78	.266	.326	.450	.776
618	166	24	8	21	73	107	.269	.346	.435	.781
428	93	14	0	7	45	85	.217	.292	.299	.591
224	48	5	1	3	18	44	.214	.273	.286	.558
156	44	7	2	2	12	26	.282	.333	.391	.724
4851	1241	220	33	94	428	890	.256	.316	.373	.689
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
455	122	19	3	11	38	84	.268	.325	.396	.720
444	101	11	3	12	38	72	.227	.288	.347	.635
219	49	10	1	3	21	42	.224	.292	.320	.611
175	33	2	1	3	11	29	.189	.237	.263	.499
382	110	25	1	14	37	48	.288	.351	.469	.819
608	168	28	7	11	32	103	.276	.312	.400	.712
440	106	13	11	9	17	103	.241	.269	.382	.651
331	78	16	3	8	21	89	.236	.281	.375	.656
462	134	14	3	19	31	91	.290	.335	.457	.791
360	72	3	3	7	21	82	.200	.244	.283	.527
378	96	9	2	17	30	90	.254	.309	.423	.732
395	88	17	1	5	36	96	.223	.288	.309	.597
127	31	1	0	3	16	28	.244	.329	.323	.652
4776	1188	168	39	122	349	957	.249	.300	.377	.677
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
404	95	17	2	8	30	91	.235	.288	.347	.635
397	108	20	2	4	41	101	.272	.340	.363	.703
439	124	30	2	8	33	103	.282	.333	.415	.747
539	169	26	2	7	43	114	.314	.364	.408	.772
649	199	37	7	0	48	171	.307	.354	.385	.740
366	92	14	6	2	20	91	.251	.290	.339	.629
294	71	21	1	3	29	68	.241	.310	.350	.660
359	82	14	1	6	34	84	.228	.295	.323	.618
406	107	19	3	7	30	98	.264	.314	.377	.691
615	184	24	17	11	65	123	.299	.366	.447	.813
522	116	22	12	12	44	109	.222	.283	.379	.662
307	55	15	3	6	17	70	.179	.222	.306	.528
111	30	7	2	3	8	32	.270	.319	.450	.770
5408	1432	266	60	77	442	1255	.265	.320	.379	.699
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
415	113	16	7	8	18	80	.272	.303	.402	.705
536	159	26	11	9	31	120	.297	.335	.437	.772
497	117	19	5	15	27	130	.235	.275	.384	.659
413	102	16	8	9	35	92	.247	.306	.390	.696
327	72	13	2	5	18	82	.220	.261	.318	.579
246	48	10	4	4	11	62	.195	.230	.317	.547
162	31	6	2	2	13	51	.191	.251	.290	.542
425	135	24	5	15	25	71	.318	.356	.504	.859
656	183	27	7	14	53	125	.279	.333	.405	.738
521	135	33	4	9	38	95	.259	.309	.390	.699
379	88	16	3	6	30	80	.232	.289	.338	.626
407	113	26	2	12	43	85	.278	.347	.440	.786
239	49	11	0	4	20	39	.205	.266	.301	.568
5223	1345	243	60	112	362	1112	.258	.306	.391	.697
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
282	53	10	0	3	31	65	.188	.268	.255	.524
276	73	10	4	6	24	52	.264	.323	.395	.718
505	139	27	12	7	28	93	.275	.313	.418	.731
554	153	34	19	12	21	108	.276	.303	.471	.774
653	186	39	21	20	35	121	.285	.321	.501	.822
512	123	16	2	4	26	110	.240	.277	.303	.580
225	47	6	1	5	9	39	.209	.239	.311	.550
344	94	6	6	11	24	73	.273	.321	.422	.742
442	115	21	6	6	34	84	.260	.313	.376	.689
507	126	30	3	13	39	103	.249	.302	.396	.699
358	82	17	0	5	26	82	.229	.281	.318	.600
430	115	20	5	13	37	80	.267	.325	.428	.753
280	82	21	0	8	57	58	.293	.412	.454	.866
5368	1388	257	79	113	391	1068	.259	.309	.399	.708
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
329	74	16	5	1	24	60	.225	.278	.313	.591
217	46	10	0	4	12	44	.212	.253	.313	.567
244	51	18	2	0	12	55	.209	.246	.299	.545
171	41	9	0	1	11	25	.240	.286	.310	.596
235	58	8	1	0	14	39	.247	.289	.289	.579
349	98	17	4	2	19	71	.281	.318	.370	.688
243	48	9	2	2	11	61	.198	.232	.276	.508
394	120	30	2	10	35	78	.305	.361	.467	.828
646	188	52	7	9	74	119	.291	.364	.435	.799
672	212	61	7	21	48	121	.315	.361	.521	.882
669	196	32	12	14	48	145	.293	.340	.439	.780
566	143	22	11	8	51	105	.253	.314	.373	.687
167	36	5	1	4	11	31	.216	.264	.329	.593
4902	1311	289	54	76	370	954	.267	.319	.395	.714
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
495	132	25	4	20	32	99	.267	.311	.455	.766
574	152	17	1	31	46	120	.265	.319	.460	.779
502	121	11	0	17	43	109	.241	.301	.365	.665
368	90	13	2	5	37	69	.245	.314	.332	.645
401	103	16	1	9	41	78	.257	.326	.369	.695
382	87	20	1	11	21	88	.228	.268	.372	.640
243	49	8	1	3	8	51	.202	.227	.280	.507
142	28	3	1	3	14	27	.197	.269	.296	.565
307	74	6	6	10	23	52	.241	.294	.397	.691
519	145	17	0	18	54	94	.279	.347	.416	.763
540	135	17	2	14	58	94	.250	.323	.367	.689
541	163	19	3	19	44	96	.301	.354	.453	.807
320	87	11	2	1	28	64	.272	.330	.328	.659
5334	1366	183	24	161	449	1041	.256	.314	.390	.704
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
402	97	14	6	5	31	87	.241	.296	.343	.639
234	51	10	3	0	22	52	.218	.285	.286	.571
449	134	21	2	7	27	76	.298	.338	.401	.739
438	111	22	1	5	26	100	.253	.295	.342	.638
352	96	27	3	3	24	72	.273	.319	.392	.711
448	113	31	2	5	28	113	.252	.296	.364	.660
433	139	22	4	6	34	87	.321	.370	.432	.802
635	190	34	4	16	71	127	.299	.370	.441	.811
681	208	32	9	21	39	125	.305	.343	.471	.814
674	217	30	15	8	44	125	.322	.364	.447	.810
588	163	22	14	2	31	122	.277	.313	.372	.686
385	107	18	15	2	21	82	.278	.315	.418	.733
271	81	10	7	1	16	61	.299	.338	.399	.737
5990	1707	293	85	81	414	1229	.285	.331	.403	.734
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
426	120	35	2	8	33	78	.282	.333	.430	.763
650	204	41	4	17	52	114	.314	.365	.468	.832
651	196	24	4	14	62	134	.301	.362	.415	.777
555	150	17	5	7	42	101	.270	.322	.357	.678
251	55	7	4	2	16	54	.219	.266	.303	.569
280	67	10	7	4	20	71	.239	.290	.368	.658
469	131	30	8	10	13	96	.279	.299	.441	.740
645	198	45	13	17	39	129	.307	.346	.496	.843
683	202	50	11	21	37	143	.296	.332	.493	.825
549	123	32	4	13	36	112	.224	.272	.368	.640
375	94	30	0	7	27	100	.251	.301	.387	.688
367	83	19	0	5	20	68	.226	.266	.319	.585
135	35	6	1	3	12	22	.259	.320	.385	.705
6036	1658	346	63	128	409	1222	.275	.321	.417	.737
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
456	123	27	2	14	24	92	.270	.306	.430	.736
630	200	38	9	15	44	129	.317	.362	.478	.840
645	174	30	12	17	72	144	.270	.343	.433	.776
433	95	18	6	5	38	94	.219	.282	.323	.606
487	148	38	6	7	35	116	.304	.351	.450	.800
563	150	32	2	6	37	164	.266	.312	.362	.674
518	160	34	6	7	47	149	.309	.366	.438	.805
525	135	36	8	2	35	122	.257	.304	.368	.671
408	98	25	11	4	36	117	.240	.302	.385	.687
519	146	29	7	7	44	130	.281	.337	.405	.742
457	118	25	4	3	34	116	.258	.310	.350	.660
481	130	38	6	4	33	114	.270	.317	.399	.716
263	73	10	5	2	11	52	.278	.307	.376	.683
6385	1750	380	84	93	490	1539	.274	.326	.404	.729
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
411	106	16	4	5	33	86	.258	.313	.353	.666
226	39	3	2	1	19	53	.173	.237	.217	.454
188	52	5	4	3	8	30	.277	.306	.394	.700
502	141	23	10	16	28	110	.281	.319	.462	.781
669	188	23	17	19	41	159	.281	.323	.451	.774
635	163	25	16	23	44	126	.257	.305	.455	.760
504	131	19	14	11	37	104	.260	.311	.419	.729
576	159	21	13	13	37	112	.276	.320	.425	.745
657	195	27	15	18	48	134	.297	.345	.466	.810
566	154	19	8	12	38	127	.272	.318	.398	.715
372	95	19	1	4	18	73	.255	.290	.344	.634
297	68	18	2	4	23	73	.229	.284	.343	.628
178	46	14	0	2	22	40	.258	.340	.371	.711
5781	1537	232	106	131	396	1227	.266	.313	.411	.724
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
313	67	13	0	4	18	78	.214	.257	.294	.551
167	37	6	1	1	17	30	.222	.293	.287	.581
214	43	13	2	0	13	56	.201	.247	.280	.527
153	30	6	0	1	10	42	.196	.245	.255	.500
143	21	3	0	2	15	42	.147	.228	.210	.438
35	5	0	0	0	3	9	.143	.211	.143	.353
1025	203	41	3	8	76	257	.198	.253	.267	.521
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
307	67	8	1	4	32	69	.218	.292	.290	.582
175	32	5	0	1	15	32	.183	.247	.229	.476
100	21	2	1	0	4	24	.210	.240	.250	.490
582	120	15	2	5	51	125	.206	.270	.265	.535
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
422	109	10	2	15	25	90	.258	.300	.398	.698
316	69	4	5	7	20	64	.218	.265	.329	.594
279	58	5	4	8	23	54	.208	.268	.341	.609
454	123	20	11	11	30	80	.271	.316	.436	.752
525	142	26	8	11	28	120	.270	.307	.413	.721
495	127	13	9	8	22	122	.257	.288	.368	.656
346	78	6	6	13	24	76	.225	.276	.390	.666
473	119	21	5	16	45	74	.252	.317	.419	.735
363	79	15	4	8	19	68	.218	.257	.347	.604
291	78	14	2	7	4	47	.268	.278	.402	.680
250	51	7	1	6	7	48	.204	.226	.312	.538
280	62	19	5	2	14	65	.221	.259	.346	.605
196	48	9	4	4	18	37	.245	.308	.393	.701
4690	1143	169	66	116	279	945	.244	.286	.382	.668
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
542	170	21	11	12	35	100	.314	.355	.459	.815
518	122	14	7	13	36	106	.236	.285	.365	.650
267	57	19	0	4	12	72	.213	.247	.330	.577
410	115	22	11	11	24	89	.280	.320	.468	.789
624	177	25	9	10	30	94	.284	.317	.401	.717
456	116	18	3	6	24	81	.254	.292	.346	.638
370	113	16	4	10	27	64	.305	.353	.451	.804
654	214	34	6	12	56	125	.327	.380	.453	.833
650	207	28	4	7	68	125	.318	.383	.406	.789
650	212	35	3	3	68	130	.326	.390	.403	.793
624	198	37	0	7	55	114	.317	.373	.410	.783
532	155	21	5	12	48	100	.291	.350	.417	.767
296	79	12	5	2	24	67	.267	.322	.361	.683
6593	1935	302	68	109	507	1267	.293	.344	.410	.753
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
445	112	13	9	4	27	91	.252	.294	.348	.643
304	77	14	3	6	27	60	.253	.314	.378	.692
503	160	27	5	8	31	96	.318	.358	.439	.797
603	175	33	5	5	29	137	.290	.323	.386	.709
484	139	39	11	4	29	95	.287	.327	.438	.766
546	145	33	9	6	29	125	.266	.303	.392	.695
564	171	45	3	6	36	123	.303	.345	.426	.771
629	180	54	1	10	45	142	.286	.334	.423	.757
656	195	43	4	18	35	126	.297	.333	.457	.790
526	129	26	6	7	29	87	.245	.285	.357	.642
349	89	14	0	9	26	87	.255	.307	.372	.679
278	66	10	1	4	12	62	.237	.269	.324	.593
161	39	11	1	0	10	37	.242	.287	.323	.610
6048	1677	362	58	87	365	1268	.277	.318	.399	.718
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
454	122	33	3	8	30	93	.269	.314	.407	.722
587	164	41	8	12	22	110	.279	.305	.438	.743
499	127	40	4	9	27	84	.255	.293	.405	.698
346	68	28	2	6	19	67	.197	.238	.341	.579
207	45	12	1	1	8	32	.217	.247	.300	.546
305	81	16	5	0	15	66	.266	.300	.351	.651
494	153	15	19	5	25	97	.310	.343	.447	.790
567	146	21	15	8	31	118	.257	.296	.390	.686
343	81	10	6	8	13	79	.236	.264	.370	.634
278	64	14	1	8	23	60	.230	.289	.374	.663
516	143	31	7	13	43	103	.277	.333	.440	.773
417	97	12	2	2	36	91	.233	.294	.285	.579
156	44	5	0	5	11	32	.282	.329	.410	.740
5169	1335	278	73	85	303	1032	.258	.299	.390	.689
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
508	131	24	9	15	38	113	.258	.310	.429	.739
440	103	18	7	13	39	98	.234	.296	.395	.692
396	111	37	3	2	28	86	.280	.328	.404	.732
631	165	75	9	13	59	149	.261	.325	.471	.795
504	121	41	3	7	42	138	.240	.299	.375	.674
376	87	17	0	11	19	82	.231	.268	.364	.633
336	96	16	1	1	25	67	.286	.335	.348	.683
273	59	10	0	0	24	66	.216	.279	.253	.532
335	81	13	1	5	36	80	.242	.315	.331	.647
241	42	4	0	8	23	58	.174	.246	.290	.537
188	43	10	0	0	20	44	.229	.303	.282	.585
271	60	5	4	3	12	53	.221	.254	.303	.557
68	14	3	0	0	7	15	.206	.280	.250	.530
4567	1113	273	37	78	372	1049	.244	.301	.371	.672
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
465	124	25	4	14	45	89	.267	.331	.428	.759
521	129	16	6	12	70	112	.248	.337	.370	.707
306	57	11	5	0	43	75	.186	.287	.255	.541
217	56	9	0	2	19	47	.258	.318	.327	.645
427	127	16	0	4	28	94	.297	.341	.363	.704
326	83	16	0	0	36	78	.255	.329	.304	.632
416	111	21	2	6	52	102	.267	.348	.370	.718
431	105	30	0	5	36	92	.244	.302	.348	.650
233	57	5	1	0	17	52	.245	.296	.275	.571
365	102	13	2	5	24	72	.279	.324	.367	.691
420	98	9	3	9	31	79	.233	.286	.333	.619
206	44	7	1	0	20	49	.214	.283	.257	.540
175	56	7	3	2	11	32	.320	.360	.429	.789
4508	1149	185	27	59	432	973	.255	.320	.347	.667
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
417	116	13	2	8	36	81	.278	.336	.376	.712
385	95	15	0	8	34	67	.247	.308	.348	.656
549	170	33	3	8	32	86	.310	.348	.424	.772
570	153	39	3	7	26	117	.268	.300	.384	.685
269	57	11	3	4	10	59	.212	.240	.320	.560
240	53	7	2	8	7	60	.221	.243	.367	.610
165	26	6	0	1	11	39	.158	.210	.212	.422
283	63	11	2	7	18	55	.223	.269	.350	.619
226	51	6	2	0	8	41	.226	.252	.270	.522
85	15	3	0	0	6	20	.176	.231	.212	.443
3189	799	144	17	51	188	625	.251	.292	.354	.647
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
434	122	28	7	7	23	96	.281	.317	.426	.744
369	86	14	7	0	17	82	.233	.267	.309	.576
238	51	13	0	0	11	57	.214	.249	.269	.518
131	28	2	0	3	5	25	.214	.243	.298	.540
335	82	8	2	6	32	72	.245	.311	.334	.645
312	82	10	2	5	22	56	.263	.311	.356	.667
263	51	13	1	1	16	54	.194	.240	.262	.503
219	50	15	5	6	10	43	.228	.262	.425	.687
335	64	15	5	10	21	56	.191	.239	.355	.594
369	90	24	8	6	19	71	.244	.281	.401	.682
308	71	15	1	5	22	60	.231	.282	.334	.616
391	99	29	3	3	29	71	.253	.305	.366	.670
200	56	12	1	6	22	40	.280	.351	.440	.791
3904	932	198	42	58	249	783	.239	.284	.356	.640
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
338	85	14	0	4	23	77	.251	.299	.328	.628
326	77	20	3	3	17	103	.236	.274	.344	.618
337	81	9	4	7	27	86	.240	.297	.353	.650
263	53	9	1	3	25	84	.202	.271	.278	.548
177	36	6	0	3	14	30	.203	.262	.288	.550
161	34	7	0	3	10	45	.211	.257	.311	.568
234	53	9	0	2	21	37	.226	.290	.291	.581
265	59	11	1	3	31	61	.223	.304	.306	.610
251	58	11	1	3	26	50	.231	.303	.319	.622
358	94	21	0	12	36	78	.263	.330	.422	.752
549	148	32	1	9	62	118	.270	.344	.381	.724
478	122	18	3	11	53	96	.255	.330	.374	.704
219	57	13	3	4	14	40	.260	.305	.402	.707
3956	957	180	17	67	359	905	.242	.305	.347	.652
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
403	105	24	2	3	32	87	.261	.315	.352	.667
373	95	23	1	2	36	70	.255	.320	.338	.658
465	131	31	5	5	35	80	.282	.332	.402	.734
443	121	17	2	8	25	83	.273	.312	.375	.687
372	82	15	2	4	16	78	.220	.253	.304	.556
220	56	14	0	5	11	47	.255	.290	.386	.676
425	126	29	6	7	12	80	.296	.316	.442	.758
532	141	29	16	4	38	103	.265	.314	.402	.716
582	170	27	19	10	47	97	.292	.345	.455	.800
552	142	28	7	6	30	132	.257	.296	.366	.661
416	106	20	2	10	24	108	.255	.295	.385	.680
280	63	11	0	3	24	74	.225	.286	.296	.583
204	59	7	1	6	13	46	.289	.332	.422	.753
5267	1397	275	63	73	343	1085	.265	.310	.383	.693
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
386	89	12	6	10	26	100	.231	.279	.370	.650
393	101	15	4	13	31	81	.257	.311	.415	.726
413	98	17	8	5	29	108	.237	.287	.354	.641
312	79	13	2	2	30	60	.253	.319	.327	.646
223	42	8	1	0	9	59	.188	.220	.233	.453
99	21	3	1	1	4	31	.212	.243	.293	.536
1826	430	68	22	31	129	439	.235	.286	.348	.634
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
407	105	17	10	7	22	82	.258	.296	.400	.697
414	96	19	4	8	24	88	.232	.274	.355	.629
264	57	7	3	4	26	38	.216	.286	.311	.597
309	79	9	2	1	27	59	.256	.315	.307	.623
276	57	10	3	2	34	40	.207	.294	.286	.580
162	31	7	0	1	11	38	.191	.243	.253	.496
219	49	7	3	3	18	67	.224	.283	.324	.607
314	78	14	2	4	21	81	.248	.296	.344	.639
361	94	13	4	7	16	71	.260	.292	.377	.669
281	70	12	0	1	20	57	.249	.299	.302	.601
395	109	16	0	4	30	65	.276	.327	.347	.674
359	95	13	5	5	34	77	.265	.328	.370	.699
184	39	5	3	2	23	40	.212	.300	.304	.604
3945	959	149	39	49	306	803	.243	.298	.338	.635
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
324	75	8	2	3	32	66	.231	.301	.296	.597
282	71	7	2	6	19	62	.252	.299	.355	.654
240	43	8	0	3	9	43	.179	.209	.250	.459
140	30	3	1	2	4	42	.214	.236	.293	.529
994	221	26	5	14	64	215	.222	.269	.301	.570
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
324	74	22	0	1	13	58	.228	.258	.306	.564
177	35	2	1	2	6	33	.198	.224	.254	.478
226	49	15	1	8	11	47	.217	.253	.398	.651
461	131	18	0	16	20	89	.284	.314	.427	.741
428	107	15	0	10	26	86	.250	.293	.355	.648
329	76	10	0	9	22	63	.231	.279	.343	.623
321	78	17	0	9	22	63	.243	.292	.380	.672
304	68	12	1	6	14	50	.224	.258	.329	.587
234	55	4	1	2	14	51	.235	.278	.286	.565
203	45	6	1	0	22	27	.222	.298	.261	.559
293	79	10	4	1	30	52	.270	.337	.341	.679
410	103	8	3	8	52	81	.251	.335	.344	.679
227	67	9	1	8	23	49	.295	.360	.449	.809
3937	967	148	13	80	275	749	.246	.295	.351	.646
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
322	76	13	4	1	19	80	.236	.279	.311	.589
314	78	12	3	1	16	74	.248	.285	.315	.600
232	52	15	2	1	20	48	.224	.286	.319	.605
301	77	6	2	1	25	73	.256	.313	.299	.612
202	46	7	1	1	16	43	.228	.284	.287	.572
198	44	13	0	1	16	42	.222	.280	.303	.583
239	51	11	1	3	22	47	.213	.280	.305	.585
169	35	3	0	0	10	42	.207	.251	.225	.476
270	73	10	4	1	17	64	.270	.314	.348	.662
411	99	22	3	11	35	115	.241	.300	.389	.690
260	54	10	3	6	7	64	.208	.228	.338	.567
331	84	16	3	4	27	77	.254	.310	.356	.667
169	39	4	1	2	9	38	.231	.270	.302	.571
3418	808	142	27	33	239	807	.236	.286	.323	.609
AB	H	2B	3B	HR	BB	SO	Avg	OBA	Slug	OPS
452	140	12	5	3	35	94	.310	.359	.378	.738
618	180	32	4	13	57	130	.291	.351	.419	.770
356	69	13	1	3	40	74	.194	.275	.261	.536
115	17	3	2	0	16	35	.148	.252	.209	.461
342	99	25	5	2	23	60	.289	.334	.409	.744
433	120	23	4	4	22	93	.277	.312	.376	.689
379	90	23	1	4	18	78	.237	.272	.335	.607
275	67	13	2	3	7	47	.244	.262	.338	.601
240	51	11	0	6	13	60	.212	.253	.333	.586
437	125	20	1	9	33	96	.286	.336	.398	.734
435	120	17	0	4	44	101	.276	.342	.343	.685
456	135	33	0	5	32	83	.296	.342	.401	.744
294	84	14	0	4	17	65	.286	.325	.374	.699
4832	1297	239	25	60	357	1016	.268	.319	.365	.684