Hoff-POWER's MLB batting average and small sample sizes
A lot of the talk the past two days has been on "small sample sizes" and Hoff-POWER's limited ABs in the major leagues. In May Hoff had a BA of .467, in June he had a BA of .350, and in Sept. he has a BA of .400.
I don't want to belabor any of the arguments on whether or not he should be on the playoff roster, but I just wanted to make sure that some people don't use the words "small sample size" without some understanding of what they mean.
First of all, without getting into the mathematics of probability density functions, let's just say that to know someone's "true" batting average, one would theoretically need an infinite number of at-bats. Practically speaking, statisticians use something called a "95% confidence interval" to estimate when one is 95% certain that something like the batting average lies between a given range.
A good formula for all of this is that if N is the number of ABs, and the player has an observed batting average of BA, then assuming that the estimates of batting average are normally distributed then we can be 95% certain that the "real" batting average (RBA) of that player lies in the interval:
BA - 1.96*sqrt(BA*(1-BA)/N) < RBA < BA + 1.96*sqrt(BA*(1-BA)/N)
What does this mean? Well, Hoff has a current BA of .387 after 62 ABs, which means that we can be 95% certain that if he continues playing at this level, his "true" or "real" BA lies between .265 and .508. If he had more ABs, this "range" would shrink further. For example, Theriot has a BA of .302 after 502 ABs, giving a true range between .264 and .340. Ward has a BA of .216 after 97 ABs giving a range between .134 and .298.
Now, I'm not claiming that BA is an important metric for measuring a good hitter, and I'm not arguing who is the better pinch-hitter, or hitter in special situations. I'm just arguing for discounting other people's arguments on "small sample sizes" since in some sense, a lot of baseball stats can be considered to be based on small sample sizes, especially when one factors in things like lefty/righty splits, etc.
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although you're right
if you’re talking about his 5-year projections, his production would decline over the next few years… if Hoffpauir were an every day player this, I would suspect he would hit around .280, possibly lower. The only thing that would make me cry small sample size about Hoffpauir is low walk-strike out ratio. In fact, his eqa for Iowa this year was .278. Sabrmetrics aside, Hoffpauir’s true value will be seen in the post-season if he makes the roster. He is 28 years old, Lee is signed with the team for a few more years, Hoffpauir could make great trade bait in the offseason, if not, he will most likely be a pinch hitter next year for the Cubs considering Ward and Blanco won’t be in a cub uniform next season. a .250 swing in batting average in your projections is CLEARLY indicative of small-sample size. If you took Jay Bruce’s first 62 at bats, it would look fairly similar, except that Bruce hit for more power. While I don’t dislike Hoffpauir, I think he is too old of a player for me to rally behind him as a power threat for future cub’s teams. I do however, like Hoffpauir’s fly-ball ratio, and the fact that he is a lefty. Daryle Ward, while having a low batting average, knows how to take his walks, he is a very patient hitter despite sitting on the bench most of the year. I think we should look into Cedeno as possible losing a roster-spot for the playoffs, considering he is basically useless, unless theriot gets hurt.
by scarymonsters85 on Sep 26, 2008 12:50 PM CDT reply actions 0 recs
I'm not certain batting average is normally distributed.
We don’t see guys 3 SDs out of the norm with .800 or .900 BAs, do we? And we don’t see guys 3 SDs below the norm with .040 or .075 BAs (unless they are pitchers) that stay in the majors…they are fired, DFA’ed, or demoted. I’m not sure about the normal distribution of BAs, like I said.
Evey Hammond: Vi Veri Veniversum Vivus Vici. V: By the power of truth, I, while living, have conquered the universe.
by dtpollitt on Sep 26, 2008 12:59 PM CDT reply actions 0 recs
Batting average is normally distributed.
Of course, it’s normally distributed among everybody. If you picture the probability distribution of hitting talent in the US, the league mean of .260 or whatever is actually somewhere at the far right end of the tail.
It’s a bit more complicated than that – I’ve got to get ready for work, but I’ll see if I can dig up some links later for those who like the maths.
by cwyers on Sep 26, 2008 1:15 PM CDT up reply actions 0 recs
Regarding whether or not batting averages are normally distributed....
First of all, the example you gave isn’t really applicable, since I was talking about individual BA, and not about an ensemble average of all players. Over all players, the BA has to be approximately normally since it must obey the Central Limit Theorem (wikipedia is your friend, and cwyers has additional info). BTW, large SDs are almost never seen anyway…but that’s another story…
On the other hand, you are most certainly right, for an individual batter, it may not be normally distributed (if I were the batter, I would bat .000 in the NL). However, I think the argument makes little difference, since I used the simple formula for normal distributions as an example. For for other distributions the formulas are more complicated but the idea is the same: the “actual” BA converges to the “observed” BA pretty slowly, and only as the number of at-bats becomes extremely large.
Good point, though!
"I'm not much of a chemistry guy, you know. Chemistry to me is a pinch-hit double with the bases loaded"--Jim Frey, Chicago Tribune, 1985.
by zevkalman on Sep 26, 2008 1:29 PM CDT up reply actions 0 recs
It's not normally distributed among players.
Because of how playing time is distributed you can treat it that way, but it isn’t actually so. (Smarter people than me will tell you it’s really Bayesian, but you can approximate it this way.)
But there is no left end of the tail for the population of pro players (high minors and majors). The consequence is that there are many more true talent .220 hitters in the league in any given season than there are .300 hitters. The reason you can treat BA as normally distributed is because none of the true-talent .220 hitters get a significant amount of playing time.
by cwyers on Sep 26, 2008 2:06 PM CDT up reply actions 0 recs
It should be "approximately normally" distributed for players though...
Yes, I agree that over the entire ensemble of MLB players, the distribution is not exactly normal. For the Central Limit Theorem to hold, the number of players in the MLB would have to be extremely large (—> infinity) for it to be exactly normal (again, this assumes that each player is “independent” of each other which is not entirely true either) but I think it should be OK to say that it is approximately normally distributed.
When you use Bayesian analysis, you actually have to assume a given a priori probability distribution. For example, one real interesting application would be to infer someone’s major league talent given his minor league talent.
Given the distribution of minor league players, and given the distribution of the same players in the major leagues, one might be able to use Bayes’ theorem to make that kind of deduction. Since I don’t get paid to do baseball analysis, I’ll let someone do it!
"I'm not much of a chemistry guy, you know. Chemistry to me is a pinch-hit double with the bases loaded"--Jim Frey, Chicago Tribune, 1985.
by zevkalman on Sep 26, 2008 2:28 PM CDT up reply actions 0 recs
To be honest...
…I didnt read this entire post (I skimmed the math portion), though I did find what I read to be really well presented. Its refreshing to see an argument presented that is well thought out and rational, so thank you for that.
But… in the playoffs, defense is much more important than during the regular season and Micah just doesnt play good defense anywhere. He’s a natural first baseman, and Lee is obviously not going to miss any time excluding injury. Hoff CAN play right field, but he’s a butcher out there and even if he doesnt misplay a ball, his lack of range might lead to a few hits dropping in.
In a perfect world Kosuke would just start hitting again, and lately he has done just that. Lets hope he keeps it up.
One thing you learned as a Cubs fan: when you bought you ticket, you could bank on seeing the bottom of the ninth.
Joe Garagiola
by Ryan at Cubshub on Sep 26, 2008 1:11 PM CDT reply actions 0 recs
although not based in data
he has gone his first 62 AB’s with a great batting average…
if he keeps it up for 200 AB’s, he will make the all-star team
if he keeps it up for 500 AB’s, he will win a batting title
the disparity between great hitters and average hitters are so large in Baseball, one could only assume Micah’s stats will decline, and regress to the mean. Considering the difference between a .260 hitter to a .280 hitter… and then a .280 hitter to a .300 hitter… I will love to see how Hoffpauir performs next year. Ryan also made a good point, he is a defensive liability. Although I disagree with him about Kosuke… I’d like to see Dero play right and have fontenot play second.
by scarymonsters85 on Sep 26, 2008 1:16 PM CDT reply actions 0 recs
The scenarios are plentiful...
…as the Cubs could go a number of different ways. In a perfect world Fukudome will find his stroke again and hit in the postseason as he did in April.
If Kosuke doesnt start to produce, you could throw DeRo in right and go with the hottest bat at 2nd, with Fontenot the likely fill-in. Not ideal, but not the worst option in the world. Im not a big fan of DeRo’s defense in right (or left, for that matter) but he wouldnt be a defensive liability and that makes him head-and-shoulders better than Hoffpauir.
If only Hoffpauir and Pie could combine to make one amazing ball-player…
One thing you learned as a Cubs fan: when you bought you ticket, you could bank on seeing the bottom of the ninth.
Joe Garagiola
by Ryan at Cubshub on Sep 26, 2008 2:41 PM CDT up reply actions 0 recs
As far as I can tell
that stat indicates how “hot” that particular player can get in a short period of time… for instance, in the first month of the season, in 94 ABs, Theriot batted an obscene .457 … that should illustrate just how small of a sample size 100 AB’s really are. Hoffpauir only has 62… and about 70 PA. a nearly .250 percentage point swing for Hoffpauir is awful because in a course of a season, a batter’s average true swing won’t be anymore than .25 percentage points. I’ve said this before, baseball is a game of failure.
As sabermetrics become more mainstream, better stats will become available to truly value a player’s output, regardless of playing time… just like the ERA is not indicative of a relief pitcher’s performance, either is BA for a part time player such as Hoffpauir. Same thing goes for Wonder Hamster, as he mostly only comes to bat when the splits favor him… same for Johnson and Edmonds.
I know more than a few people who think Hoffpauir should be an everyday player because they see his BA, and this thread pretty much would end all debate on that.
by scarymonsters85 on Sep 26, 2008 1:27 PM CDT reply actions 0 recs
Excellent post!!
Sometimes people throw out the small sample size cliche when it suits their position on a given player and not when it’s really true or meaningful. We saw the same thing about giving Pie a longer opportunity earlier this year when he was hitting terribly and had struck out 23 times in 62 ab’s.
Answer not a fool according to his folly, lest thou also be like unto him.
by cubfever7 on Sep 26, 2008 2:27 PM CDT up reply actions 0 recs
This post is interesting...
but it just made my head explode.

Proud recipient of a hot dog shot from the Iowa Cubs hot dog gun.
by IowaCubs- on Sep 26, 2008 2:28 PM CDT reply actions 0 recs
The other thing I want to point out here...
…is that what you’re talking about is true if we know nothing else about the batter. We do know quite a bit more about Micah Hoffpauir than simply his BA and his at-bats. Even if that’s all we know about Hoffpauir himself, we know a bit more about major league ballplayers than that. I think I can say quite confidently that he does not have a true-talent BA of .508. Why? Because that’s not how the distribution of BA exists in MLB.
What we want is:
(m1/s12 + m2/s22)/(1/s12 + 1/s22)
Where m1 is player BA, s1 is player STDEV, m2 is MLB average BA, and s2 is the STDEV of BA in MLB.
by cwyers on Sep 26, 2008 2:33 PM CDT reply actions 0 recs
That's wrong...
…because SBN doesn’t like the carrat. s12 is really s1 squared, etc.
by cwyers on Sep 26, 2008 2:40 PM CDT up reply actions 0 recs
That's an equation that is often used to compare two measurements subject to two
different kinds of “noise” or variance.
The equation says that you measured Hoff’s BA as m1, and allow a given STDEV, and then take the average BA of the MLB, and use its STDEV. The result of that is an expected value. Well, given that those are two different measurements of two different things, I’m not convinced it’s valid to use it in that way….but maybe you can argue that you’re just adjusting the large STDEV1 in Hoff’s BA by the expected STDEV2 of the MLB? Maybe it’s true, but I ’m not convinced yet.
Without getting too mathematical, I’ve typically used that kind of equation to measure electromagnetic signals at two different times, when lightning was happening at one of the times, and so the noise level (STDEV1) was higher…anyway, this discussion is getting too theoretical. If you’re ever in DC, I can buy you a cold beer and you can try to convince me of its validity. :-)
"I'm not much of a chemistry guy, you know. Chemistry to me is a pinch-hit double with the bases loaded"--Jim Frey, Chicago Tribune, 1985.
by zevkalman on Sep 26, 2008 2:58 PM CDT up reply actions 0 recs
You probably know far more about this than I do!
I’m not really a “statistics” person, although I have some rudimentary familiarity with things like RMSE and regression and use them from time to time. Most of what I do is more akin to data mining or something like that.
That’s not the only equation I’ve seen for estimating true talent from observed performance, but it was the one that was the closest match to the concepts you were presenting. Most of the time what I use is a variation on what Tom Tango does in the Marcels projections, where you simply include X PAs of league average performance when figuring the weighted average.
by cwyers on Sep 26, 2008 4:04 PM CDT up reply actions 0 recs
Good post...
…plenty of room for some discussion but I think the gist is that the more AB’s a player has, the more significant his average stats become.
I posted this earlier but when I am considering small sample size stats, I ask myself how drastically would these numbers be impacted by a couple of AB’s going the other way.
For instance, if a guy has a 5 for 5 night, is this evidence of a good hitter or just a product of small sample size? To answer that I change the outcome of 2 AB’s making him 3 for 5. This is still a very good night so I’d say yes on that night he was a very good hitter.
No let’s say a guy has gone 4 for 19 against a certain pitcher (hitting .211). Not so good right? But if you change the outcome of 2 AB’s he is 6 for 19 (.316). So I would say that the sample size isn’t sufficient to make a judgement of whether this is a good or bad judgement. Obviously, you can’t throw all of this matchup data out the window but you can assign some level of confidence to certain matchups. You should assign the most confidence to matchups where there is lots of data or very one-sided data.
"I'm petrified of nipple chafing. Once it starts, it's a vicious circle." Andy Bernard
by TXCub on Sep 26, 2008 2:59 PM CDT reply actions 0 recs
Let's keep in mind
that he started at least some of those games. What are his stats SOLELY as a Pinch Hitter? Hoff seems to be the kind of guy who does well when he gets a lot of looks at a pitcher, not just one in a high-pressure clutch situation.
"That’s the great thing about baseball, you never know what’s going to happen till you get the final out." — Lou Piniella
by drewishdrewid on Sep 26, 2008 3:39 PM CDT reply actions 0 recs
again, you're
using a very small sample size. His pinch hitting compared to his starting statistics are irrelevant because he only has 62 at bats. as a pinch hitter though, he is 3-13
by scarymonsters85 on Sep 26, 2008 6:46 PM CDT up reply actions 0 recs

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