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## Thursday, August 27, 2009

#### Freakonomics Blog:  Statistical Slumps

It occurred to me that it would be pretty easy to derive a statistical standard for determining when an athlete was having a “statistically significant slump.” For example, Alex Rodriguez recently went through a homerless drought of 72 at-bats. Over his career, A-Rod has averaged one homer for every 14.2 at bats — suggesting there is about a 93 percent chance that he will not homer on any individual at bat. It would be crazy to say that he was in a home-run slump after failing to homer after just a few at bats. But the question is how many homer-less at bats is enough to be a statistically significant drought?

The answer is 42. There is less than a 5 percent chance that Rodriguez would go homerless 42 times in a row — so we can reject the hypothesis (at a 5 percent level of statistical significance) that he is going homer-less merely as a matter of chance. You can calculate your own drought statistics for any sporting event (for example, how many losses does Tiger have to have before he’s having a statistically significant drought?) just by using the following formula:

Athlete is having a statistical significant drought if:

Total consecutive number of bad events > log(.05)/log(probability of single bad event)

Dr. I likes his panda steak medium rare Posted: August 27, 2009 at 01:15 AM | 39 comment(s) Login to Bookmark
Tags: sabermetrics

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1. joker24 Posted: August 27, 2009 at 04:06 AM (#3306404)
Uhhh no. Given a random sample alright, but it's very much not randomly selected. You're specifically looking at that sample because it is "special". Alex Rodriguez has had 400 PAs this year, don't feel like doing the actual math, but I'd guarantee 42 homerless ABs is not significant at all.
2. The Most Interesting Man In The World Posted: August 27, 2009 at 04:12 AM (#3306408)
The answer is 42.

I didn't realize that the question would be included.
3. Blackadder Posted: August 27, 2009 at 04:13 AM (#3306410)
I actually don't remember what the right way to calculate the probability of ARod having a 72 PA homerless streak given his career home run rate. The issue is that he has a lot of 72 PA stretches which overlap, and hence are very much not independent. I have a vague recollection from my analysis class that you can use the Doob Stopping Time Theorem to calculate the expected number of PA ARod would need before having a homerless streak of any given length, but that isn't exactly the question I'm asking.
4.  Posted: August 27, 2009 at 04:37 AM (#3306417)
He homers in 6% of his plate appearances, so the likelihood of no homers in x PA = 1-.94^x
5. Avoid Running At All Times- S. Paige Posted: August 27, 2009 at 04:48 AM (#3306420)
The answer is 42.

I didn't realize that the question would be included.

Really, I think this is the only reason this was written by the author of this piece. I loved that book as a youth.
6. Blackadder Posted: August 27, 2009 at 05:22 AM (#3306427)
That's in a fixed collection of x PA. I am wondering about the probability of his having some streak that long at some point in his career, not in any one fixed set.
7. Lassus Posted: August 27, 2009 at 06:30 AM (#3306445)
I lost that bet, as I would have put the over/under and 1.5 posts and taken the under.

But snark beat meta-nerd reference by six minutes.
8. tjm1 Posted: August 27, 2009 at 07:22 AM (#3306456)
You need to lower your probability for calling it significant a lot. A 5% chance means 1 in 20. There are about 300 major leaguers who get a lot of at bats every year. About 15 of them should do something that happens only 5% of the time per year. But these homerless streaks, for most players, will appear to be significant if they less a small fraction of the year. So you need a multiple for that, too. I'd guess you want to get that probability down to 1 in 1000 or less before worrying about it. There's also Blackadder's point, but I think the trials issue is probably a bigger one.
9. Biff, highly-regarded young guy Posted: August 27, 2009 at 08:34 AM (#3306459)
I loved that book as a youth.

Why only as a youth?
10. Dr. I likes his panda steak medium rare Posted: August 27, 2009 at 10:03 AM (#3306462)
I actually don't remember what the right way to calculate the probability of ARod having a 72 PA homerless streak given his career home run rate. The issue is that he has a lot of 72 PA stretches which overlap, and hence are very much not independent. I have a vague recollection from my analysis class that you can use the Doob Stopping Time Theorem to calculate the expected number of PA ARod would need before having a homerless streak of any given length, but that isn't exactly the question I'm asking.

I don't have enough probability theory training to answer this question analytically.

While it is not the analytical approach, I would expect Monte Carlo methods would take care of this problem pretty quickly. It would be a simple matter of generating a lot of "careers" of homerun/no homrun trials for ARod, and then looking at statistics around homerun droughts. Of course, this wouldn't really tell you much about how you would actually expect these streaks to unfold (a real baseball player probably doesn't have a constant homerun rate for his career), but it would certainly answer this particular question.

While TFA seams off in its calculation, I found the author's notion of statistical significance as a tool for sportswriters kind of funny.
11. Avoid Running At All Times- S. Paige Posted: August 27, 2009 at 10:23 AM (#3306464)
Why only as a youth?

Good point. I do not have a way with words. But I guess I never read the Hitchiker series as an adult. I'm sure I'd still love it though.
12. bookbook Posted: August 27, 2009 at 11:12 AM (#3306470)
I have read them as an adult. Like Heinlein and Mad comics, it's just not the same.
13. sunnyday2 Posted: August 27, 2009 at 11:25 AM (#3306472)
I don't think age has anything to do with it. I first read the Hitchhiker as an adult and loved it. It's reading it the second time, at any age, that is the problem. The Lord of the Rings, however, still works the second time. What are the odds of that?
14. Rally Posted: August 27, 2009 at 12:00 PM (#3306477)
How many HR chances does A-Rod get if he comes to bat 6 times per game in a 9 game stretch?
15.  Posted: August 27, 2009 at 01:39 PM (#3306515)
This also doesn't account for the fact that A-Rod is not 100% healthy this year and that over any small sample his chances of hitting a HR are going to be affected by the game situations and pitchers he faces.
16. Vogon Poet Posted: August 27, 2009 at 01:51 PM (#3306531)
I don't think age has anything to do with it. I first read the Hitchhiker as an adult and loved it. It's reading it the second time, at any age, that is the problem.

Why do you say that? I've read the Hitchhiker's trilogy four or five times and still love it.

The upcoming sixth book worries me, however.
17. JoeD has the Imperial March Stuck in His Head Posted: August 27, 2009 at 01:55 PM (#3306538)
You need to lower your probability for calling it significant a lot. A 5% chance means 1 in 20. There are about 300 major leaguers who get a lot of at bats every year. About 15 of them should do something that happens only 5% of the time per year. But these homerless streaks, for most players, will appear to be significant if they less a small fraction of the year. So you need a multiple for that, too. I'd guess you want to get that probability down to 1 in 1000 or less before worrying about it. There's also Blackadder's point, but I think the trials issue is probably a bigger one.

I don't understand how this applies.

I didn't RTFA, but if the author is saying a 42 AB streak is statistically significant, all he's saying is that it isn't necessarily random at this point, and that perhaps something is 'up' here, like maybe he's hurt, maybe his girlfriend dumped him, his uncle died, etc..

Everything isn't random. I think he's saying after 42 AB you should be concerned that he isn't 'normal' as there's a less than 1 in 20 chance this would happen randomly. I think that's entirely reasonable. Players aren't the same random APBA or Strat-O-Matic card every day of the season.
18. tjm1 Posted: August 27, 2009 at 02:05 PM (#3306550)
Everything isn't random. I think he's saying after 42 AB you should be concerned that he isn't 'normal' as there's a less than 1 in 20 chance this would happen randomly. I think that's entirely reasonable. Players aren't the same random APBA or Strat-O-Matic card every day of the season.

My point is that 42 at bats is about 1/15 of a season. A guy like ARod should have a 42 AB homerless streak every season or two, if everything's random.

Look at it another way: Say I'm playing blackjack. As the player I win just under half the time, but let's call it half the time, to make the math easier. I have a 1/16 chance of winning four hands in a row. But if I play 100 hands, I'd expect that somewhere in there, I'd win four in a row. It wouldn't mean that the odds had suddenly gone in my favor or anything. It's just that unlikely events happen more often when you do the experiment more times.
19. tjm1 Posted: August 27, 2009 at 02:10 PM (#3306559)
By the way, I'm not saying that ARod's homerless streak is definitely a random fluke. What I'm saying is that it certainly could be a random fluke, and that just random flukes alone are enough to produce a 70 at bat homerless streak for one of the top 20 or so sluggers in the majors about once every year or two.
20. JoeD has the Imperial March Stuck in His Head Posted: August 27, 2009 at 03:07 PM (#3306603)
My point is that 42 at bats is about 1/15 of a season. A guy like ARod should have a 42 AB homerless streak every season or two, if everything's random.

I understand that . . . but what I'm arguing is that maybe it isn't random, and that there is a reason for it. And if you don't start having some concern for it at that point (if you are the player or the hitting coach) perhaps it will continue past 42 AB.

I think it's reasonable to say that if you're down to the point where it's 19:1 that it would be random, there might be something going on there.

I would say that you could argue that it isn't random, that every year or two a slugger has something so wrong that he ends up with a long streak.

Since we are comparing it to standards that include these 'random' events (they are incorporated into the numbers we use to determine how often these things should occur) they may appear to be random, when quite possibly, they aren't.
21. tjm1 Posted: August 27, 2009 at 03:49 PM (#3306660)

I would say that you could argue that it isn't random, that every year or two a slugger has something so wrong that he ends up with a long streak.

Well, of course, none of this is really random. Either the guy hits the ball fairly squarely, near the sweet spot of the bat, or he doesn't. His margin for error depends on the pitch speed, the ballpark, the weather, and his bat speed.

My point is that if you calculate what the probability is that you have a random event, on the basis of the guy's rate of home runs per at bat, and you expect that number to mean something, then you should take into account all the batters in the league and all the stretches of at bats for each batter.

And even if you take out a 42 at bat homerless stretch of a typical ARod season, you only increase his home rate from about 7% of at bats to about 7.5%. It doesn't make much of a difference to the calculation. It would drop the threshold to about 38 at bats for a 5% chance of being random, if viewed in isolation.

I understand that . . . but what I'm arguing is that maybe it isn't random, and that there is a reason for it. And if you don't start having some concern for it at that point (if you are the player or the hitting coach) perhaps it will continue past 42 AB.

What I would say is that looking at the home run rate, rather than watching the consistency of his swing, and seeing how frequently he made fairly solid contact, would be the wrong approach, because home runs are so rare. Let's remember, 42 at bats is only about a week and a half for an everyday player.
22. kthejoker Posted: August 27, 2009 at 03:52 PM (#3306665)
tjm: You're confusing the events. The event is not "every 42 AB", it's "every set of AB between A-Rod home runs."

95% of these events < 42.

That doesn't imply causation or anything, but it also means you can't use 42 AB = 1/15 season to determine how often the event occurs.

EDIT: Or to put it another way, to measure it the way you want to measure it: from 2007 to present, ARod has 111 HRs in 1426 ABs, which is almost exactly 34 "42 AB" chunks. That's basically 3 HRs ever 42 ABs. Calculate the probability that over any given "chunk" A-Rod will hit zero home runs (poisson, e^-3 * (3^0) / (0!) = .0497870684.)
23. Harmon "Thread Killer" Microbrew Posted: August 27, 2009 at 03:58 PM (#3306675)
This is a trick question, as it doesn't tell us who the opposition's middle infielders are in these scenarios.
24. tjm1 Posted: August 27, 2009 at 04:02 PM (#3306681)
EDIT: Or to put it another way, to measure it the way you want to measure it: from 2007 to present, ARod has 111 HRs in 1426 ABs, which is almost exactly 34 "42 AB" chunks. That's basically 3 HRs ever 42 ABs. Calculate the probability that over any given "chunk" A-Rod will hit zero home runs.

Yes, that's not a bad way to do it. Assuming a Poisson distribution, the chance of getting zero when you expect 3 is also about 5%. So my original points hold.
25. tjm1 Posted: August 27, 2009 at 04:04 PM (#3306685)
Anyways, the exact answer is going to depend on exactly what question you ask. But the point is that if there are many opportunities for something to happen, which there are here, and the chance that it will happen in any one of those is 5%, then it's not actually terribly surprising if it does happen.

And the point on when a coach should become concerned is that the coach shouldn't be looking at statistics of relatively rare events to make these determinations.
26. Blackadder Posted: August 27, 2009 at 04:17 PM (#3306710)
No, if he has had 1426 AB, then he has had 1384 chunks of 42 AB. It's just that a lot of them overlap, and hence are not independent, which is precisely why this is not trivial to compute.
27. tjm1 Posted: August 27, 2009 at 04:41 PM (#3306740)
No, if he has had 1426 AB, then he has had 1384 chunks of 42 AB. It's just that a lot of them overlap, and hence are not independent, which is precisely why this is not trivial to compute.

But the chance of any given 42 at bat chunk having no home runs is trivial to compute, if you assume that the home runs are Poisson distributed with a single rate over all the at bats.
28.  Posted: August 27, 2009 at 05:21 PM (#3306783)
For most of this, we should really be using binomial, not Poisson. Homers are binary for our purposes and we're assuming an exact probability of 1 HR every 14 AB. There's no possibility of A-Rod hitting 2 home runs in a single at-bat, after all, so the likelihood of 42 AB without a home run is 4.45% not 4.98%.
29. tjm1 Posted: August 27, 2009 at 05:22 PM (#3306786)
For most of this, we should really be using binomial, not Poisson.

Yeah, but once the numbers are large enough, it's nearly the same. There are far more troubling issues than that one.
30.  Posted: August 27, 2009 at 05:42 PM (#3306820)
I can't do it because I don't have the proper tools on my laptop, but I'm with Dr. I. Monte Carlo is plenty accurate for the purposes we're talking about when the prospects of having to otherwise whip out the OMG TEH STOCHASTIC CALCULUS seem to be good.
31. tjm1 Posted: August 27, 2009 at 06:00 PM (#3306853)
OK. I did a quick simulation. For 10000 at bats, 7% of at bats being homers, the longest homerless stretch was 126 at bats. For the first 600, the longest homerless stretch was 45. For the next 600, it was 71. Then 45, then 47, then 78, then 83, then 68.

Basically, though, the point is that it is not out of the ordinary at all to have a stretch of 70 or so at bats without a homer, even if you average 40 homers a year.
32. Maxwn Posted: August 27, 2009 at 06:17 PM (#3306873)
I think the problem is in the premise of the question. I think it would be correct to say that the probability that Alex Rodriguez will not hit a homerun in his next 42 at-bats is less than 5%, like TFA calculated. However, once you have already observed 42 homerless at-bats, TFA's calculation is not the answer to the question "How improbable was it for Alex Rodriguez to not hit a home-run for 42 at-bats?" That's a totally different question. The first question deals with one pre-specificed period, the second deals with a large number of periods and deals with the probability that one of them meets the criteria.
33. JoeD has the Imperial March Stuck in His Head Posted: August 27, 2009 at 07:00 PM (#3306922)
My point is that if you calculate what the probability is that you have a random event, on the basis of the guy's rate of home runs per at bat,

Bingo. That's the problem. ARod is not always 7% or 1:14 to hit a HR. Even if you leave out things he has no control over like pitcher and park, etc..

When everything is clicking, he's 100% healthy and his head is on straight, it might be 11%. If his hip is bothering him, or his hitting mechanics are slightly off, or his ex-wife is being a pain in the ass, or Jeter doesn't help him out in a tough spot with the media, or he made an error last inning and he's still thinking about it, he might be 3%. It's a moving target that averages out at 7%.

So just saying that he hasn't HR'd in 42 AB and you'd expect that doesn't cut it, it's the easy way out.

It could be random, sure. Or it could be that he's going through some of the issues that have moved his true level down from 7% temporarily. I would tend to put my money on the latter.
34.  Posted: August 27, 2009 at 07:16 PM (#3306940)
I think Joe in #33 is exactly right. Nothing that happens on a baseball field is random in the sense that dice baseball is random, even if dice baseball simulates the results well.

In fact, even without outside distracting factors, the "moving target" average concept would seem to be a feature of all head-to-head competitive sports. AROD is sitting on fastballs, hitting 11% home runs, and word gets around and guys start throwing him junk on the outside corner. AROD is thrown off, starts hitting 3% home runs, then adjusts and starts hammering the junk, so he's back to 11% again and starts to see high fastballs. (That's a gross oversimplification of real-life pitching patterns, of course, but it approximates the adjustments and counter-adjustments of competition.)

I suspect that there's really "something wrong" with a player even when a slump is fairly short; it's not just "random" unless you can easily see that he's hitting the ball square but the defense has truly robbed him, that kind of thing. But, all else (like injuries and ex-wives) equal, the better players adjust more quickly, and their brief slumps combine with their successes to produce a more consistent pattern of success.
35. tjm1 Posted: August 27, 2009 at 07:35 PM (#3306960)
It could be random, sure. Or it could be that he's going through some of the issues that have moved his true level down from 7% temporarily. I would tend to put my money on the latter.

I did the simulation with a homer in 11% of his at bats. Even then, in 10000 at bats, there were 7 stretches where with more than 42 at bats between homers. Since 10000 at bats is about 17 seasons worth, you'd expect to see one of those stretches per player every 2 or 3 years.

So, for a 42 at bat stretch without a homer to mean anything, the level a guy has to be at when everything is going right has to be extremely high.

I think a good coach/scout could probably see something wrong in the player's approach over 42 at bats. I think the statistics have a hard time showing anything in a 42 at bat stretch.

I'm not disagreeing with you about any of the details about all these calculations being too simple. However, it's pretty clear to me that if you want to read a lot into a 42 at bat stretch, you need to have a very high home run rate.

Another way to look at it is the chance of hitting three homers in a game. For an 11% rate, you'd expect a guy to do this every 50 or so games in which he has 5 at bats. You'd expect a 2 homer game every two 5-at bat games with an 11% rate. If a player were really have stretches of higher rates, and stretches of lower rates, you'd expect a lot more 2 and 3 home runs games than you actually get from guys who hit 40 homers a year. Now if the fluctuations are at-bat to at bat, then they don't have very much impact on long streaks. They'll have some, to be sure, but they'll only matter a lot if you start having very large deviations from the average.

EDIT:

Only every 6 games for a 2 homer game, not every 2.
36. tjm1 Posted: August 27, 2009 at 07:37 PM (#3306963)
I suspect that there's really "something wrong" with a player even when a slump is fairly short; it's not just "random" unless you can easily see that he's hitting the ball square but the defense has truly robbed him, that kind of thing. But, all else (like injuries and ex-wives) equal, the better players adjust more quickly, and their brief slumps combine with their successes to produce a more consistent pattern of success.

Maybe, but the question is whether you should be using the statistics to judge when a player is struggling, or the other information you have available. Also, should you use the statistics of rare events like homers, or more common things like base hits as a whole?
37. JoeD has the Imperial March Stuck in His Head Posted: August 27, 2009 at 08:46 PM (#3307035)
Like most things, I would say a combination of both.

As fans, most of us can't just watch every game, so we naturally tend more towards the statistics side. If I were a coach and able to watch every at bat, I'd use both. Or maybe I'd say, you know what, ARod hasn't homered in 2 weeks, maybe I should focus a little more on him (time is a limited resource and you have 12-13 other guys to worry about as well), and see if there's something wrong. Etc.
38. tjm1 Posted: August 27, 2009 at 09:02 PM (#3307055)
If you're a coach, you not only watch every game, but you also probably have video from three different angles, bat speed information, and all sorts of fancy software to look at the swings. I'm guessing that the good hitting coaches can just watch two or three swings, and tell that something's off, and can figure it out on the video in 20 minutes. A lot of the good hitters watch the video themselves, and know how to break it down - Gwynn was famous for this, of course, but I bet ARod does this, too.
39. Dr. I likes his panda steak medium rare Posted: August 27, 2009 at 10:13 PM (#3307127)
Linked in the comments section of TFA. It appears to be an applet that calculates streak probabilities. I haven't looked it over that closely yet, and it is dinner time, so it might be a while before I get to it.

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