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Ray Allen’s Hot Streak

Posted by Neil Paine on June 7, 2010

In case you missed last night's game, Ray Allen went crazy from beyond the arc in the first half, knocking down his first seven 3-point shots of the contest. As Henry Abbott notes, Allen is a career 40% shooter from beyond the arc, on 6678 career attempts. So Henry asks:

"If you hit 40% of the time, and take 6,678 shots, how often would you end up with seven or more makes in a row?

Does Allen do that more often than you'd expect? (Bring on your probabilities!) If the answer is yes, then let's talk about the hot hand. But if the answer is no, well then let's appreciate this is the kind of night good shooters have sometimes, even without the supernatural."

Let's address the first question -- on average, how many runs of 7 straight 3-pointers would you expect a Ray Allen-like career (40% shooter, 6678 shots) to contain? And what's the probability that he would do it at least once in his career? Well, there's no easy way to set this up in equation form, because you have to account for the possibility of multiple sequences containing at least 7 consecutive makes, which would require some heavy-duty combinatorics. Instead, when confronted with a problem like this, I like to set up a Monte Carlo simulation and derive the probabilities by running a large number of trials (for other examples of posts where I did this, see here, here, here, here, and... well, you get the idea).

With the experiment set up to simulate five thousand 6,678-shot "careers" with a 40% chance of making each shot, we find that (on average) each career contains this many streaks that ended at each length:

Streak Instances
0 4007.7
1 961.9
2 384.7
3 153.6
4 61.6
5 24.5
6 9.8
7 3.9
8 1.6
9 0.6
10+ 0.4

So if you hit 40% of the time and take 6,678 shots, you can expect to have about 6-7 stretches (6.5, technically) in your entire career where you have seven or more makes in a row. Also, you'll have at least one stretch like that in your career 99.9137% of the time -- in essence, this means a "true" 40% shooter is virtually guaranteed to have at least one run like Allen's in his career due to chance alone. These stretches can come in one half, one game, or even across multiple games; in fact, we find that the best streaks of all time (Brent Price & Terry Mills in 1996, not coincidentally when the arc was shorter) made their 13 straight across several days. But how many times has Allen rattled off at least 7 in a row during his career?

Unfortunately, it's impossible to answer that question with our current dataset; you would need a play-by-play log of every Allen game to determine exactly how many times Allen has had a streak of each length. But if anything, it seems like it happens less than you'd expect due to chance -- if you're a true 40% shooter with a career as long as Ray Allen's, a stretch like last night's should happen roughly every other season. But in real life the odds of making your 7th in a string of 7 straight are probably less than your odds of making the 1st, simply because the defense will have made some adjustment to stop you from doing what you're doing. This would mean they were under the belief the Hot Hand exists (otherwise, they'd be just as content to take their chances with 40% on the 7th attempt as they were on the 1st).

Also working to drag down the chances of making 7 straight in real life is the tendency for players who make multiple shots in a row to force subsequent shots in the belief they were hot, only to miss those "heat-check" attempts at a higher rate than normal. In other words, Ray Allen the 'true" 40% shooter might think he was on a run that boosted his true ability to, say, 50%, so he takes a shot he normally makes only 30% of the time on the next possession... but quickly realizes that it's still a 30% proposition. Only when Ray stays within his normal shot selection is he a "true" 40% shooter, and that shot selection can be changed by his own choices, as well as those of the defense.

Either way, though, we can say with some certainty that a streak like Allen had last night is indeed possible due to random chance alone. While it's unlikely that any given stretch of seven consecutive 3-point attempts would produce 7 makes (the probability is 0.4^7, or 0.1638%), if given enough stretches of seven consecutive shots, you're inevitably going to see some runs where he does in fact make all 7. That it happened in an NBA Finals game -- and not a meaningless Celtics-Nets game in February -- was all the more beneficial for Boston, but Allen's streak neither proves nor disproves the Hot Hand... It was simply a great performance by one of the most skilled 3-point marksmen in the game's history.

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27 Responses to “Ray Allen’s Hot Streak”

  1. Mike G Says:

    When an all-time great shooter has just hit a few, the defense is going to react to quell the hot streak. If he continues to hit them, he's actually overcoming much greater odds than normal.

    Since all players are aware of hot streaks -- having had them, getting the ball to hot teammates, and having been burned by hot opponents -- not one player on the floor is unaware of who is hot, ever.

    It's possible that the hot shooter will then undermine his hotness by taking shots he shouldn't be taking. Of course, some of the time he'll make even those, because he is hot. More certainly, he'll get fewer good looks because the defense is rushing to put out the fire.

    It's amazing to think that some still think basketball players are the only living things that might not have extra-focused times when they are particularly accurate.

    Henry Abbott really thinks it's either "this is the kind of night good shooters have sometimes", or else it's "supernatural"?
    It's the most natural phenomenon there is.

  2. AYC Says:

    No offense, but when someone says there's no such thing as a "hot hand" it sounds like they have never played the game. I know my own anecdotal experience isn't going to convince anyone of anything, but I'd just like to point out a logical flaw in the anti-hot-hand argument: players aren't machines, or roulette tables. They THINK about the shots they attempt (well, except Artest); shooting a basketball is a mental, as well as physical act. It's simple-minded to attribute hot-streaks to mere "randomness" when human agency is involved. Do the naysayers think "basketball IQ" doesn't exist either?

  3. Neil Paine Says:

    I've played. I've thought I had the hot hand. I've come to believe I was wrong about my own experiences when it comes to "the zone"... Although that's not to say far greater players don't actually have real "hot hand" abilities. I would hope you would have gleaned from this post that I'm a "Hot Hand Agnostic" -- I can't prove it exists, I can't prove it doesn't exist, and I don't think either fact will ever change.

  4. AYC Says:

    That wasn't a shot at you, Neil. I appreciate your agnosticism on the matter. I was responding more to Abbot, I guess....

    But what about my main point, about the mental aspect of playing/shooting? There's nothing "supernatural" about the effect of the Mind. Were players like MJ, Magic and Bird great entirely because they were "more skilled", or was part of their greatness due to the fact that they were SMART players? Being a great shooter/scorer may have a lot to do with muscle memory, quickness and size, but it also has to with reading the defense and shot-selection. By way of another example, consider Allen Iverson, who was a skilled player but not very smart, or Bill Russell who was smart but not very skilled (offensively)

  5. Neil Paine Says:

    I'm not really sure why belief in basketball IQ requires you to believe in the hot hand, though. My understanding of the hot hand (and I could be wrong) is that sometimes players inexplicably are capable of playing beyond their true skill level for some amount of time. And when I say "skill level", I mean their physical and mental skills (the ability to read defenses and understand where to be on the court in relation to where your teammates are is certainly a skill).

    I've always heard "the zone" or the hot hand (are they the same, btw, or is it wrong to use the terms interchangeably?) be given almost mythical properties, basically like clutch hitting in baseball. When it comes to clutch, I won't try to argue whether or not it's possible for a player to consistently perform below his skill level in pressure situations -- although it strikes me as unlikely since pro athletes have been groomed since childhood for pressure situations, and a defect in that area would likely have manifested itself and weeded them out long before they made it to the highest level.

    But I must protest the idea that a player can consistently perform better than his base skills are capable of, simply because it's a pressure situation. What is the mechanism by which this happens? If the player can do it by choice, why doesn't he do it all the time? The same goes for the zone -- how is it possible for a player to increase his base skill set in the middle of a game? What is the mechanism by which this occurs? It can't be "focusing more", because that would imply the player was capable of going into the zone at will, which seems to go against the definition of the phenomenon.

    I guess I just have doubts about how a player can increase his base skill level mid-game. A player can certainly play better than he's capable of due to matchups or even random chance, but his skill level hasn't changed. The hot hand and the zone say that a player's core skill level actually increases.

  6. AYC Says:

    Consider the star player in a two-for-one situation at the end of a quarter; say he shoots a quick shot and misses, then runs down the clock and hits the buzzer-beater after getting the ball back. Did he hit one and miss the other because of the law of averages? No, both times he made the smart play based on an understanding of the situation and the clock. A lesser player might have taken more time to get the first shot, and cost his team the last possession; but if he hit that shot, as far as the stat-head is concerned player A is 1-2, while player B is 1-1 (more "efficient"). All these contextual matters and the thought processes of players are completely ignored by the anti-hot-hand crowd.

  7. Neil Paine Says:

    Right, but that can be accounted for if you knew how much time was left in the quarter -- a study could discount shots like that and only look at situations where the clock wasn't a factor. I don't know if they've done this or not, but I think they're talking about not being able to string together consecutive makes in normal clock situations as well.

  8. Jason J Says:

    Even disbelief in a hot hand doesn't mean that a defense wouldn't adjust to a play call that was creating open 3s for a shooter as good as Allen. Nobody wants to take their chances with an open Ray Allen at any distance. That's just common sense. Let's start jumping the passing lane (like Kobe did which led to a great fade to the corner catch and trey by Allen) or jump the shooter with the screener's man (as Bynum did to force the airball at the end of the clock).

  9. Brian Says:

    Count me as another basketball fan who both values statistical analysis tremendously, and also believes the "hot hand" is obviously true from personal playing experience.

    Neil wrote:
    "My understanding of the hot hand (and I could be wrong) is that sometimes players inexplicably are capable of playing beyond their true skill level for some amount of time."

    I think it's more like this: there is no such thing as a permanent, unchanging "true skill level". There is a skill level that tends to be in a certain region of "skill space", but fluctuates over time. As Mike says, it is not mythical at all to believe in fluctuation in human performance. Some days you feel sharp on the job, and other days you just don't have it.

    Same is true in basketball. For some stretches of time, a basketball player may have the intricate mechanical nuances of his jump shot just right. At other times, the player may experience some slight hitch in his mechanics that takes some time and/or effort to fix. Sometimes these fluctuations are very strong. We call the crest of a strong fluctuation in skill level "being in the zone" and we call the trough "being in a slump."

    Can focusing help you perform better? Yes, just like in any other area of life. Why not focus all the time then to maximize performance? Ask why you aren't 100% focused in any other activity. Focus uses up mental resources; why not sprint as hard as you can whenever you're in an NBA game? If you can focus better in important game situations, why not focus better all the time? Well, focus is not an entirely self-generated thing. Certain contexts lend themselves better to focusing. I bet Michael Jordan couldn't focus himself for a pre-season game as well as he could for an NBA finals game no matter how hard he tried.

    What is the mechanism for fluctuations in performance? One possibility is spontaneous fluctuations in brain activity that have been scientifically demonstrated to vary with fluctuations in motor task performance, e.g.:

    Neuron. 2007 Oct 4;56(1):171-84.
    Intrinsic fluctuations within cortical systems account for intertrial variability in human behavior.

    Fox MD, Snyder AZ, Vincent JL, Raichle ME.

    Abstract

    The resting brain is not silent, but exhibits organized fluctuations in neuronal activity even in the absence of tasks or stimuli. This intrinsic brain activity persists during task performance and contributes to variability in evoked brain responses. What is unknown is if this intrinsic activity also contributes to variability in behavior. In the current fMRI study, we identify a relationship between human brain activity in the left somatomotor cortex and spontaneous trial-to-trial variability in button press force. We then demonstrate that 74% of this brain-behavior relationship is attributable to ongoing fluctuations in intrinsic activity similar to those observed during resting fixation. In addition to establishing a functional and behavioral significance of intrinsic brain activity, these results lend new insight into the origins of variability in human behavior.

  10. AYC Says:

    I feel like all this talk of the "Zone" being almost "supernatural" or "mythical" is something of a straw-man, one that worked it's way into your definition of what being in the Zone means. It's not about a player mystically playing beyond their abilities, it's merely about a player playing to his full capabilities for an extended period. A great shooter like Ray Allen can easily hit 50 threes in a row in the gym; put in that context, hitting 7 in a row during a game is hardly miraculous.

    And obviously a player can't play at peak level all the time; most players become more mentally focused in the 4th, or at other critical junctures.

  11. Brian Says:

    I also think there is individual variability in how much skill level tends to fluctuate over time. Some players are relatively steady and others are relatively streaky. I consider myself a very streaky jump shooter. There is nothing mystical about it, it's just variation in mechanics. When I'm hot, my mental command "take jump shot" executes all the right mechanics. When I'm not hot, the same mental command "take jump shot" executes mechanics that are flawed. At that point, one needs to pay attention to mechanics to figure out what is going wrong.

    There is a give and take between effort and effortlessness. Optimum performance takes place when you feel like you're doing things automatically (c.f. "he's unconscious!") and not paying a lot of attention to the mechanics. But when you're not paying a lot of attention to the mechanics there is danger that the correspondence between your high-level intentions ("take jump shot") and low level motor execution strays from the optimum. At that point you need to engage effort to fix the mechanics, and then hopefully get back to the point where effortful attention to the mechanics can be abandoned.

    As a Knicks fan I've seem some players that subjectively seem very steady in their jump shooting performance (Allan Houston) and others that are very streaky (John Starks, Jamal Crawford). Check out this video of Crawford, for instance, scoring 52 points on the Heat, including one stretch where he made 16 consecutive field goals.

    http://www.youtube.com/watch?v=DxKEou4vPHk

    That's a player in the zone, not random chance.

  12. Mike G Says:

    Last night, Ray Allen was "extraordinary" in the first half. There was something 'extra' beyond his 'ordinary'.

    If a player is being "supernatural", that's beyond what occurs in nature. Lots of phenomena were called this, before they were understood.

    Doctor J made an extraordinary move to score from behind the backboard in the 1980 Finals. As far as film archive can tell, he never did it before or since. It wasn't supernatural, though. He might only make it one time in six.

    We don't debate whether it was supernatural or mystical, because we've all seen it, and in slow motion. He was at max ability on that play.

    Those who believe in Hot Hand via superstition are as wrong as those who debunk it on the same grounds. Of course, some just Believe in Kobe; that he will 'deliver' them, etc. He probably believes it. A couple of tough shots earlier may lead to a bunch of bad ones later.

  13. Neil Paine Says:

    Like I said, I'm an agnostic. A lot of the explanations listed above could be completely true... or they could be an attempt after the fact to assign some kind of rational reason for a process that's governed by random variation. The jury is still out right now.

  14. marparker Says:

    Ah yes the hot hand,
    I hear about the hot hand alot at roulette tables and craps tables.

  15. Ryan. Says:

    I've played basketball my entire life. To me, there is no question here. The hot hand exists whether you can quantify it or not.

    Sometimes, you're just on a different level. Your mind and body alike go to a place they're simply not usually at.

  16. DSMok1 Says:

    It is a very interesting situation that BECAUSE people believe the hot hand exists, it will be obscured in the data--purely because of A) defenses focusing and B) players forcing. Basically, teams and players apply game theory as if the hot hand exists.

    If the hot hand actually exists but game theory adjustments compensate, there will be no evidence of the hot hand in the stats. If game theory adjustments overcompensate, there will actually be "negative" evidence of the hot hand.

    In fact, I would say that if there is no evidence of the hot hand (i.e. the player is just as likely to make the 8th shot in a row as the 1st) then we should take that as evidence of a hot hand, because it is well known that teams are adjusting as if there is a hot hand. Does that make sense?

  17. DSMok1 Says:

    It's not like baseball, where there aren't really any easy "game theory" adjustments to attack "clutchness". The hot hand can be quickly be adjusted for by adjusting defenses.

    I believe the hot hand exists, personally.

  18. Neil Paine Says:

    Well, that would just mean the null hypothesis was that the hot hand exists, and the burden of proof should be on the numbers to prove it doesn't. I'm not saying this is wrong, I'm just saying that's what a statement like "if there is no evidence of the hot hand, then we should take that as evidence of a hot hand" means. If the null hypothesis was that there wasn't a hot hand, then under the same logic, the lack of evidence that there wasn't a hot hand would be evidence that there wasn't a hot hand. It all depends on what the default assumption is -- and we've all seen the conventional wisdom's default assumption be wrong before.

  19. DSMok1 Says:

    Yeah. How does one measure for it, though, if the defense is likely to immediately adjust for it?

  20. Mike G Says:

    Does anyone doubt that, for example, golfers have massive hot and cold spells?
    There's no defense in golf.

    An opposite extreme is the playground basketball skirmish where everyone is his own 'team'. Jim gets hot and goes a couple of baskets ahead; the game (for everyone else) becomes "stop Jim".

    If Jim is hot enough, he'll win on shots beyond his normal range. If it's random luck, he'll be shut down quickly.

    Nick Anderson, a 70% FT shooter, went 0-4 at the end of a Finals game. This will happen one time in 80. Then he went on to shoot 40% for a whole season. Then 64%. Without defense.

  21. Joe Says:

    How many times has Ray Allen 'missed' eight threes in a row.

  22. Ryan. Says:

    Beat me to it, Joe.

    Jesus,Ray (no pun intended), way to play the ultimate turnaround.

  23. AYC Says:

    What is the supernatural cause of the mysterious "Ice Cold Hand" I wonder?

  24. Ryan. Says:

    Why are we using the term "supernatural" to begin with?

    Athletes can increase their vertical leap on any given jump at any given time due to VARIOUS factors, of which some are mental above physical. This has been documented multiple times in sports analysis and is and echoed by pro-ball players often.

  25. Brian Says:

    See also this important paper, which critiques the null hypothesis testing approach to evaluating the hot hand. Using formal model comparison, the author found that models positing variation in "true" skill level fit observed basketball data better than models than posit one static "true" skill level.

    http://tinyurl.com/38spklr

    Detecting the Hot Hand: An Alternative Model
    Yanlong Sun (Yanlong.Sun@uth.tmc.edu)

    Abstract
    The belief in the hot hand was suggested to be a “cognitive
    illusion” since no significant evidence was found in the
    basketball-shooting data to reject the simple binomial model
    (Gilovich, Vallone & Tversky, 1985). The present study
    argues that in order to evaluate the validity of human
    perception and cognition such as the hot hand belief, a datadriven
    approach is needed to compare multiple alternative
    models. A hot hand model with nonstationary shooting
    accuracy was tested and showed significantly better
    approximation to the data than the binomial model, indicating
    that the simple binomial model may not be accurate enough to
    serve as a normative model. This finding suggests that the hot
    hand might indeed have existed, and weakens the argument
    that the hot hand belief might be “seeing patterns out of
    randomness.”

  26. P Middy Says:

    The Law of Averages is a stone cold beotch.

    Game 2, Ray goes for 8 for 11 from 3.
    Game 3, Ray goes for 0 for 8 from 3.

    8 for 19 = 42.1%

    His playoff average? 41.6%

  27. Onu Says:

    Been thinking about this a lot recently, and I’m increasingly of the opinion that the issue lies with an understanding of what hot hand means.

    First, lets be clear that ‘average’ means exactly what it says. It is not the representation of any given game, but a collective assessment over a wide range of games. The actual experience in any given game of that average is virtually always going to be different from ‘the average’- apart from that fluky game where a player shoots his exact average!

    In that context, I think that what ‘hot hand’ means is that in this particular game, a player’s scoring mechanics are most finely aligned (tendons and ligments, mental game, e.t.c) while in a ’slump’ they are most off. So in game 2, for some reason, Ray’s shooting mechanics were about as sharp as they can be, while in game 3, they were as off as they can be. In game 4, they are more likely to return to an intermediate state, in which the actual manifestation is more in line with what has been seen before (5/7, 3/8, 2/5 shooting nights, that kind of thing).

    The issue I see with statistical statements that say a certain sequence is bound to happen even randomly is that it obscures what is going in the mind and body of a player, which is not entirely random at the point of experience. Rather I believe what should actually be indicated is that for any given good basketball player, there is a very high likelihood that playing mechanics will be so finely tuned that an exceptional game will result at some point, and that we should not be surprised if that happens.

    The surprise comes if that exceptional occurence happens for so long that it appears to be a performance even beyond the rare exceptional performance expected from good players. At this point, we could maybe call it the 'miraculous hand' and make a much bigger deal out of it :) Thus to my mind, it is entirely okay to believe in the 'hot hand' and note those performances, as long as hot hand means the rare occasions when a players entire playing tools (mind and body) are entirely in sync. Perhaps Neil you could do a spread analysis on the Ray's shooting percentages so we can see the most exceptional shooting games (or streaks of games?) he's had (good and bad) and we can term his experience in those games 'hot hand/streak' or 'slump'.