Sorry about all the sports talk lately, but the NBA playoffs are afoot and I have basketball fever.  I’ll get back to physics soon, I promise.

There’s been a lot of talk lately in the “geeky basketball fans” community about whether the “hot hand” exists in basketball.  If a player has been making most of his recent shots, is he more likely to make the next ones?  Or is it safe to assume that his recent success is just a matter of statistical fluctuation?  In other words, should players/coaches try to “feed the guy with the hot hand”, or is it always better to go with the player with a higher career shooting percentage?

I, personally, believe in the existence of the hot hand.  The main reason is this: I once made 25 consecutive free throws.  At the time I was only about a 50 percent shooter (I’m only marginally better today), and I had just finished a rough game of church-league basketball.  Before going home I decided to take fifty free throws, and somewhere in the middle I made 25 in a  row.  If you think that those 50 shots were all statistically independent of each other, then the odds of me, a 50% shooter, making 25 consecutive free throws somewhere in the middle of a set of 50 are literally one in a million (about 0.75 in a million, actually).

The hot hand, in my experience, is better described as a “hot brain”.  When I’m “hot”, I feel like making a shot is just a matter of concentration.  It’s like I remember, perfectly, the sequence of muscle firings required to make a shot.  So if I just concentrate hard enough on the shot, I’ll make it.  Conversely, when I’m “cold”, it’s like I can’t remember how to shoot correctly anymore.  Some shots go in, but I can tell that it was luck.  My mind does not have the right memory loaded up.

So I believe in the hot hand.  And if it can happen for me, I was fairly confident that it must happen for professional basketball players as well.  I was pretty sure that I had seen it, too.  After all, I was watching when Tracy McGrady scored 13 points in 35 seconds.  In the interview afterwards, he said “the basket felt as big as the ocean to me.”

This weekend, I searched for the hot hand in two places.  The first was in shot-by-shot data from the NBA.  The second was at the local gym, where my wife and I took two hundred shots and recorded the sequence of outcomes in an attempt to create that statistically improbable streak that would provide evidence for the hot hand.

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The NBA

Ryan at basketballgeek.com has compiled wonderful data sets for every play from every game in the last three NBA seasons.  I decided to test the data from last season, 2007-2008, in a few different ways.  The first was to define a correlation function, the standard technique in physics, of a given player’s shots.  What I hoped to find was some player for whom a shot was influenced by the one that preceded it.  In other words, I wanted a player whose misses and makes tended to group together in a way that was statistically unlikely.  I most certainly did not find this.  Every player I could think to check (Kobe Bryant, Ray Allen, Dwight Howard, Ben Gordon, Kyle Korver, Dwyane Wade, and a few more ) showed absolutely no significant correlation between the outcome of one shot and the outcome of the subsequent ones.  Previous studies have shown that there is a slight negative correlation between a shot and the one that immediately follows it, which has been interpreted as an “overconfidence” problem.  i.e. when a player makes a shot he is more likely to take a bad shot the next time he has the chance.

My second method was to look for correlations between a player’s shooting percentage at the beginning of the game and their shooting percentage at the end of the game.  For example: if Kobe Bryant opens the game shooting 8/10, how many of the next 10 shots is he likely to make?  Should you expect him to shoot close to 80%, or close to his season average of 45.6%?  The answer, resoundingly, is that his season field goal percentage is a much better predictor than his shooting percentage in the first 10 shots.  But is there any correlation at all between early-game and late-game shooting percentage?  Maybe a small one…  that depends on whether you trust that these data points really do have an upward slope:

Kobe's shooting percentage: first 10 shots vs. subsequent shots. Do you believe the trend line?

Finally, I searched through data from dozens of shooters for streaks of consecutive shots in order to find a highly improbable sequence.  Was there some equivalent among the NBA ranks of my 25-in-a-row performance?  The answer, surprisingly, was no.  Of course there were some unlikely sequences, but none so unlikely that you shouldn’t have expected them to happen over the course of a season.  For example, Kobe had a streak of 7 consecutive makes on Nov. 4, 2007 against the Utah Jazz.  The odds that any seven of Kobe’s shots would all go in is about 0.4%, and I’m sure that any observer in the building would have said that Kobe had the hot hand at the time.  But Kobe Bryant took 1,602 shots that season.  The odds that there would be a streak of 7 consecutive makes somewhere during the season is almost 70%.  So the fact that Kobe had this streak of 7 is not at all inconsistent with the hypothesis that every shot is uncorrelated from the ones before it.  Every NBA shooter that I examined gave me similar results.  Sure, there were some improbable streaks, like Ben Gordon making 8 in a row, but nothing so unlikely as to be inconsistent with random variation.

So, in the end, I find no real evidence for the hot hand in the NBA.  That doesn’t mean it doesn’t exist, just that people’s shot patterns would look the same if it didn’t.  The results are perhaps best summarized by a quote I found in this marvelous little document on sports streaks and the stock market: “Long streaks are, and must be, a matter of extraordinary luck imposed on great skill.”  It’s well worth reading.  Apparently a Nobel laureate in physics once examined data in baseball and found that the only streak which could not be explained by random variation was Joe DiMaggio’s “one in a million” 56-game hitting streak.

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The gym

With my NBA analysis rebuked, I went to the gym with my wife to take some shots and collect some data.  The goal was to do what no NBA player had been able to do: provide irrefutable evidence for the hot hand.  We decided to take 200 shots each: 100 free throws and 100 randomly-distributed shots from about 15-foot range.  Each time we had one person shooting and the other rebounding/recording the results.  For the randomly-distributed shots, the rebounder also “randomly” chose the location of the next shot.  We were careful to spend a roughly equal amount of time on each shot.  These came fairly rapid-fire, with no breaks in between.  The results are as follows, where 1 represents a make and 0 a miss:

I’ll save you from counting and tell you that my wife beat me 65 – 61 in free throws and 57 – 54 in the random shots.  Highlighted in yellow are three statistically unlikely streaks.  The first is a streak of 13 consecutive during my free throw shooting.  And I have to say, I definitely felt “hot” during that streak, like my mind and body were on the same page.  Statistically, though, the odds of a 61% shooter making at least 13 consecutive somewhere during a 100-shot sequence is 23%.  So a statistician might say that my streak of 13 was unlikely, but hardly irrefutable evidence.  Similarly, my streak of 11 makes during the random shots had a probability of about 19.7%  of happening, given my 54% rate of making those shots.  Taken together, you might conclude that I’m a little bit of a streaky shooter.  This data is hardly going to rock the basketball world, though.

The really strong evidence is in Mrs. Gravity and Levity’s streak of 10 misses during the random shots.  She shot 57% in that set, so the probability of her missing 10 straight was very small.  In fact, the probability of a 57% shooter missing 10 consecutive somewhere during a sequence of 100 is only 5.8%.  So for her to miss 10 in a row is pretty strong evidence of something going wrong — she had a “cold hand” during that stretch.

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So what do we take away from this?  The shooting patterns of my wife and I show some evidence for the hot hand, and stronger evidence for the “cold hand”.  But why didn’t this show up in the NBA data?

My conclusion is that really good shooting is a matter of concentration.  When you’re shooting free throws alone in a gym, it’s relatively easy to get into “the zone”, where your concentration level is exactly where it needs to be to make long streaks of shots.  But in a real game, where defenses are doing everything they can to break your concentration, “the zone” is a much harder place to reach and maintain.  I will probably continue to believe that Kobe Bryant “had the hot hand” last night when he torched my beloved Utah Jazz with ridiculous, hand-in-the-face, turnaround jumpshots, because being a basketball fan is largely about belief.  But now I will do so with the awareness that his shot patterns would look the same even if there were no such thing.

April 26, 2009 3:58 pm

I’ve spent a fair amount of time lately thinking about the”hot hand,” especially since my data was used (somewhat to my chagrin) to demonstrate the “cold hand.” Like most people, I have had moments where I felt like I was shooting much better than what is normal for me. However, I wouldn’t describe it the same way as this post does. For me, it isn’t about concentration or focus at all. When I have the hot hand, I feel as though I can shoot without effort or concentration and make it every time. It’s as though there’s some bodily instinct that knows exactly what to do and how to do it without much input from me.

As far as detecting (or, rather not detecting) the hot hand in NBA statistics, I would suggest an intrinsic difficulty in methodology. The idea of the hot hand is that once you get it, you are much more likely to make subsequent shots than your season or career average would suggest. So it makes more sense to correlate the average of shots made at the beginning of a purported hot hand episode with subsequent shots, rather than trying to correlate the first ten shots with subsequent shots. The latter method assumes that a shooter has the hot hand from the very beginning of a game until its conclusion. However, if you try to correlate the average at a beginning of a streak with subsequent shots you start to get yourself into a circular argument: you have to find a hot handed streak first, then calculate the correlation between the early shots and subsequent shots, all in order to demonstrate that the streak existed in the first place. So, then, I would think that searching for the hot hand in NBA player statistics is destined to be a fruitless endeavor. But, that doesn’t mean the hot hand doesn’t exist. I, for one, would like to think that if I’m capable of the cold hand, I should also be capable of the hot hand. At least once in a while.

2. April 29, 2009 7:34 pm

What initially brought me here was your post about the fastest possible mile. It is interesting stuff and was also discussed in “The Lore of Running” but I think the ultimate mile was a little bit faster – maybe 3:30.

On Dimaggio’s hitting streak – I don’t have the math in front of me, but a few years ago I read an article that theorized that Ichiro Suzuki had about a 3% chance of breaking the record. He hit for a high average, rarely walked, and played on a team that scored a lot of runs. These things all helped his possibility of having a long hitting streak.

Good stuff – I’ll keep reading!

April 29, 2009 8:44 pm

Thanks Brian; I’m glad you like it. If you have references to these things, feel free to post them. The more information we have, the better.

3. May 3, 2009 8:49 am

The ultimate mile information came from “The Lore of Running, 4th Edition” by Dr. Tim Noakes. He summarized three different studies on predicting ultimate human performance. The first methodology used a linear prediction (done in 1974) and has shown to be overly optimistic based on the times since then. No ultimate performance was presented because this would eventually say you get there in 0 seconds! The second method used an approach similar to what you presented. The ultimate prediction for the mile is 3:28.40. The third method used three physiological principles and how these are improving over time. It predicts an ultimate mile of 3:18.87. These actually talk about all times from 100 meters to the marathon so it is pretty interesting stuff.

My baseball books are currently in a huge pile that are somewhat unreachable due to disarray in my library. Too many books, not enough bookshelves! I’ll keep looking for that study.

May 3, 2009 1:39 pm

“The goal of physics is to predict the future.” — Is it now? That statement seems a little too broad, doesn’t it?

May 3, 2009 1:44 pm

The above comment was in reply to the Skee-ball post.

May 4, 2009 1:48 pm

I said it in sort of a silly way, I suppose, but it is my genuine opinion.

You can argue that the goal of science is to understand the present, but if you ask me, our main goal in science is to gain predictive power. We want to know: what will happen if I do ___ ? With predictive power comes the ability to control our fate a little bit better.

May 15, 2009 12:09 am

The statistics on hot hand are interesting however, I think you have missed a very important class of players in the NBA which are most likely to demonstrate this trend (the streaky shooters). By focusing on the NBAs stars the incredible number of shots in a year make most events and streaks somewhat probable. However, consider a player like Eddie House from the Boston Celtics. In the game last week May 6th he went 11-14 from the field with 8 consecutive field goals. This has to be especially improbable considering he is only shooting 0.445 from the field with 560 attempts on the season. In addition, the same player went 6-7 the following night as well as, 5-5 on May 2nd, and countless other games with unlikely hot streaks. Eddie House also went 0-7 April 14th, and 1-10 on Nov. 1st which I would guess would also be very improbable streaks for such few shots. I don’t have the full database of field goals to easily filter, but would guess that there are a variety of players in the NBA with substantially less attempts than the superstars, but are far better evidence of hot and cold hands.

May 26, 2010 2:48 am

I’m 45 and still play twice a week. I’m usually the second or third top scorer on the floor in our 35 and older pickup group. I run like crazy, pass well and can never predict from one week to another how I’ll play. Also – and maybe this is true for you – at this point in life warmup shots usually have little correlation to how I’ll do in a game. FYI I’m not a big 3 point shooter, I might take 2 a month. More midrange stuff in my game.

I’ve noticed over the past few years that some things stand out when I’m on a streak:

1 It’s all about form. If I’m nailing shots – including once going for 6 straight in a game – everything feels perfect. Elbow at the right angle, release is perfect, rotation is just so. Usually – seriously – someone else compliments me on my form on these nights. “your jumper is just perfect tonight!”

2 The first shot- well, it’s nice if it goes in, no biggie if it does not. If I miss shots #2 and #3, though, I start to worry if I missed #1. If the missing streak continues I tend to shut down the scoring, set picks, rebound and pass. If I was being checked for a shooting average that night it would just kill the average – several misses followed by almost no other shots. Does that happen in the NBA? Probably not to Kobe, Pierce or LeBron too often. They keep jacking up the shots, but it’s not uncommon to hear that they were 10-26.

If the first shot goes in I wonder “hmmm….”. If #2 AND #3 go in, I figure I’m hot. Then, given my experience (I’m 45, after all) I’ve learned to pick my shots and not go crazy. Unless my team needs me to score I’ll pass and stack up the assists. If I’m hot I get double teamed, hence someone is open, usually for an easy layup in our group. Assists lead to even hotter shooting in my book. Drives up the average – fewer shots, more makes, more assists.

3 Usually a bad shooting night = clangs. Not near misses, not “that should have gone in”. Clunk, clang, thunk. Either it’s in the middle of the net (or a pure bank, something the young guys don’t get much yet, I bank maybe 1/3 of my shots) or a long rebound. To me banks have a 25% better chance of dropping.

4 I also do remember those “in the zone, garbage can sized hoop” events. Can’t miss. Jacking up threes, not usually my shot, knowing, just knowing, even before I go up that not only is it going in but it’s going in the middle of the net and the net is going to flop up and need to be pulled down somehow. Knowing that if my defender gives me a half second or looks away for a moment I can make it no matter where I’m at.

5 Finally, I’d like to see some sabermetrics done on shooters from the past. Larry Legend, Dominique, Tripucka, Microwave, Scottie, Isaiah, etc. I think shooters were better in the 80’s and earlier as fundamentals were stressed more. If Larry missed 4 in a row (did he ever?) was he more likely to hit #5? Kobe can’t hold a candle to Larry when it comes to shooting. Also, compare jumpers and perhaps layups, not dunks. LeBron is a great athlete and can dunk nearly whenever he wants. Compare his jumper stats to Tripucka’s or even Majerle’s jumper stats/flow/streaks and see what comes out of the math.

8. April 14, 2011 12:30 pm

Very interesting post, as I’ve seen papers published in stats journals that did a much poorer job of this analysis and often done with just awful data. It seems that people who tend to do these studies, unlike you, don’t actually play and thus have no real feel for the game. I’ve always wondered about multi-point correlation functions and the “hot hand”: if a player hits two in a row, is he more likely to hit the third, or three in a row and the fourth, and so on?
I’m a firm believer, by the way. I’m about an 80% free throw shooter (in practice) and once had a streak of 33 (naturally, there was no one else in the gym at the time). I’m quite a poor 3 point shooter (maybe 20% from the college line) and once hit 11 in a row. Like someone else in the comments though, I think one should select for streak shooters, and not great players or even high percentage shooters. Vinnie the “Microwave” Johnson and “Downtown” Freddie Brown come to mind from the past. Hard to think of many today though. Perhaps Steph Curry?