Saturday, June 6, 2009

The Infamous Fisher "0.4" Shot


Perhaps no playoff shot has been dissected, debated, or deconstructed as much as the "0.4 Shot" made by Derek Fisher in Game 5 of the 2004 Western Conference Semifinals between the Los Angeles Lakers and the San Antonio Spurs. The Lakers did not win the title that year (they went on to be defeated by the Detroit Pistons in five games), but the closeness of the timing and the marquee nature of the two teams, who had combined to win the last five championships, conspired to focus unprecedented attention on the game-ending jumper.

Much speculation centered around whether Fisher could humanly have caught the ball, turned around, and released the ball, all in the 0.4 seconds available to him. Spurs partisans insisted that he couldn't possibly have done all of those things in so short a time; Lakers fans responded that Fisher didn't do all of those things sequentially, but combined them so that he could do them all. My own personal impression (possibly colored by my bias as a Lakers fan) was that the clock started somewhat late, but not substantially so.

Fortunately, there's no need to rely on anything so nebulous as whether Fisher's shot was plausible or not. Missing from all these speculations was an examination of the actual footage. Video from the game captures instants of the game that, for the live angle at least, are equally spaced in time. The video can therefore be used as a kind of "clock" to determine the interval of time that Fisher had possession of the ball. In assembling this particular look at the Fisher shot, I used a video file that was encoded at 25 frames per second (as I determined by stepping through frames at the end of each quarter). Unfortunately, this was not the native frame rate of the original broadcast, and this increases the random error involved in timing intervals between events. It should not, however, produce any systematic bias one way or the other.

By figuring out how many frames pass between the time that Fisher catches the ball and the time he releases it, and dividing by 25 frames per second, the elapsed time can be calculated. The bottom line, for those who are impatient or don't care about analysis: About five to six tenths of a second elapsed between the time that Fisher caught the ball and the time that he released it.

The Game

On May 13, 2004, the San Antonio Spurs played host to the Los Angeles Lakers in Game 5 of the Western Conference Semifinals. After leading most of the game by as many as 16 points, the Lakers went cold from the outside while the Spurs came steadily back, eventually going ahead 71-68 on a layup by Tony Parker with a little more than two minutes left in regulation.

After a timeout, Shaquille O'Neal responded with a turnaround eight-foot jumper in the lane to bring the Lakers to within a point. The teams traded empty possessions until Kobe Bryant sank a 19-footer from the left wing on a screen by Karl Malone, putting the Lakers ahead 72-71 with 11.5 seconds remaining.

After a non-shooting foul by Derek Fisher, the Spurs inbounded the ball in their frontcourt with 5.4 seconds left. Manu Ginobili passed the ball into Tim Duncan, and tried to cut to the basket for a return pass, but got tangled up with O'Neal and was out of the play. With no other clear options, Duncan faked one way, then dribbled the other toward the top of the key, taking a blind fadeaway jumper that touched nothing but net. The clock read just 0.4 seconds.

The Lakers called timeout. Dejected and weary players trudged slowly back to the bench, none wearier than Bryant, who was exhausted not only by the 47 minutes he had played in the game, but also by the constant jetting back and forth between the team and his legal troubles in Colorado. The Lakers' play out of the timeout called for the players to stand in a stack near the top of the key, in an attempt to break out one of their stars, O'Neal or Bryant, for a quick shot or a tip-in. But before the Lakers could inbound the ball, the Spurs called a timeout. They had seen enough, they hoped, in order to defend the play well.

After the timeout, the Lakers came out in a different set, with the players scattered across the halfcourt. Players cut, especially Bryant, but with Robert Horry doubling on Bryant rather than playing Payton inbounding the ball, Payton couldn't find an open teammate and had to call the Lakers' final timeout.

When the ball was brought into play for the final time, the Lakers returned to their original set. Bryant broke out from the stack toward halfcourt, tailed by Horry and Devin Brown. O'Neal curled toward the basket, while Malone drifted toward the top of the key. Finally, Fisher broke toward Payton.

Payton tossed the ball, leading Fisher toward a spot about 18 feet from the basket on the left wing. Fisher began angling his body for the turn before catching the ball in mid-air, then coiled on his legs and prepared to shoot over Ginobili's outstretched arms. At seemingly the same instant, Fisher released the ball, the game horn sounded, and the backboard's red light came on. Nineteen thousand people held their collective breath. The ball arced upward and came down; Fisher thought he had pushed it off too hard, but it was offset just enough by his backward motion from the basket, and the ball fell perfectly through.

A hush fell over the crowd as Fisher ran down the court in celebration, eluding his mobbing teammates and streaming down the tunnel toward the locker rooms. Rasho Nesterovic and Kevin Willis waved their hands to indicate the shot got off late. Duncan stood unmoving, hoping they were right. The referees, who had called the shot good when it happened live, convened at the scorer's table to examine the video of the play from the ABC cameras. A few tense minutes passed before the referees confirmed their initial call was correct: The shot was good. The Lakers had won Game 5, 74-73, and returned home to trounce the Spurs in Game 6 to win the series, 4-2.

The Aftermath

Writing on May 14, the day after Game 5, Dusty Garza, the editor of Spurs Report, relayed news that the Spurs had filed a formal protest with the league office, claiming that the clock started too late after Fisher touched the ball, and that the shot should not have counted. The league denied the protest that same day.

Garza also offered his personal opinion that Fisher's shot did not get off in time—indeed, could not have gotten off in time—based on the notion that human reaction time is, on average, three-fourths of a second (750 milliseconds). Since the clock couldn't have started any faster than that, Garza wrote, Fisher could have had anywhere up to a bit more than a second to shoot the ball.

This seems an unreasonable conclusion. In the first place, Garza contends that the average human reaction time is three-fourths of a second, then says that unless the referees are superhuman, they couldn't possibly have pushed the button less than three-fourths of a second after Fisher touched the ball. Well, if the three-fourths of a second is an average, wouldn't half the human population be able to do it faster (assuming negligible skew in the distribution)? And presumably NBA referees are trained to be a bit faster than average.

Secondly, research indicates (Laming 1968, Welford 1980) that simple reaction time—the time required to do something simple like push a button after a visual stimulus—is more like one-fifth of a second (200 milliseconds), rather than three-fourths.

What's more, it's unclear that human reaction time is involved here at all. Bennett Salvatore once said, speaking to Henry Abbott of ESPN's TrueHoop blog, that NBA referees don't anticipate calls; they only observe the game. However, that can't possibly be literally true all the time. When Payton passes the ball in-bounds, it is immediately evident that the ball will be caught (or at least touched) first by Fisher. It is human nature to anticipate this first contact, and act accordingly. But what does it mean to "act accordingly"?

For years, the clock was operated manually, by the timekeeper, based on the rules of the game and the whistles of the referees. The system worked well most of the time, but placed a lot of reliance on the alertness of the timekeeper.

In 1999, the NBA installed a new system developed by Mike Costabile, an NCAA referee who previously officiated in the NBA. Each referee carries a small transmitter attached to his or her belt, with a button. When the clock is to start, each referee pushes the button at the exact instant at which he or she believes the ball to be in play. The first button push triggers an automatic start to the clock. The system also includes a microphone that is sensitive to the particular frequency of the whistles used by NBA referees, and stops the clock when the whistle is blown.

In order to activate the clock, at least one of the referees must push a button at the instant he or she believes the ball to be first touched. Obviously, this generally doesn't happen right at the moment the clock is "supposed" to start. There are two potential delays here: reaction time, and execution time (the time it takes the finger to actually push the button).

To understand the relationship between these two, and how the actual delay is affected by context, suppose I ask you to clap your hands as soon as one of the following happens:

  • A basketball I drop from four feet hits the floor.
  • I clap my hands.
  • I move my hands at all.

In the first case, it takes about half a second for a basketball released from four feet up to hit the floor. That is enough time for you to react and execute the act of clapping your hands at the precise moment the ball hits the floor. In the second case, both of us need our execution times to clap our hands, but you have to react to the start of my clapping motion. In counting the delay, your execution time is cancelled out by my execution time, leaving just your reaction time. And in the last case, I can move my hands without warning, meaning that the delay is your reaction time plus your execution time.

In the case of the play in question, it took about half a second for the ball to pass from Payton's hands to Fisher's hands. For a referee who is ordinarily alert, this is plenty of time to predict the path of the ball and press the button almost immediately upon contact. Even if we accept that referees do not anticipate events, they must at least be prepared for the potential event of contact between Fisher and the ball; there is no reason, at any rate, for the delay to be anywhere near three-fourths of a second.

But let's not be too hard on Dusty Garza. He was writing in the heat of the moment, and from honest feeling. Besides, let any of us without team favoritism cast the first stone. Let's get down to brass tacks: Garza sincerely believes that the video shows that Fisher took about a second to get the shot off. Did he?

The Video

The video file I used in putting this article together is encoded at 25 frames per second. (I determined this by advancing the video frame by frame, 200 frames, at the end of each of the four quarters, when the clock is counting down at tenths of a second. Each time, 200 frames corresponded to an interval of exactly 8 seconds, so the video must be progressing at 200/8 = 25 frames per second.) Therefore, each frame represents 1/25 = 0.04 second. This is not the frame rate of the original broadcast, which would probably have been 30 frames per second. As part of the re-encoding process, frames would lose some definition, increasing the error involved in estimating precisely when events happen. Since these errors do not accumulate, however, they add a small random error, but they do not systematically bias estimates of interval lengths.

One problem in reviewing this particular game video is that only the live shot actually keeps time accurately. In all subsequent replays, the ABC crew slowed the video at a variable rate, in order to allow Al Michaels and Doc Rivers to comment on it. But the live shot has the camera in line with Fisher and Payton, making the determination of the instant Fisher first caught the ball difficult. Here, for instance, are three successive frames of the live shot, obtained using the snapshot function of the xine video player. Which one of these frames do you think shows Fisher actually catching the ball?

I think it's pretty clear that this angle can't be used to reliably determine when Fisher touched the ball (which would have started the clock). Fortunately, we can still make use of other camera angles. This has precedent in the NFL, in which "composite" video reviews are conducted. This allows the referee (and video replay official) to assess multiple angles in order to come to a firm conclusion, even when no single angle provides all of the information necessary. This isn't to say that the NFL uses fancy three-dimensional visualization tools (à la The Matrix), since they can't do that in time, and neither is it necessary here. We'll just combine the angles mentally.

Here are video stills from the opposite baseline. It shows the ball and Fisher approaching one another. In this first frame, it's not very easy to see the ball, but you can see it superimposed on referee Joe Forte's right foot. At this point, the ball is still a couple of feet from Fisher's outstretched hands.

The frame below shows Fisher and the ball considerably closer to one another, but there still appear to be several inches in between them.

This third frame shows Fisher's hands possibly touching the ball for the first time. They don't clearly touch, but this is the first frame from this angle where contact is plausible.

Note the positions of the other players. The positioning of the left foot of Duncan (guarding Malone at the free-throw line) is especially revealing. That foot covers the free-throw line, as seen from this camera angle. Duncan's foot must therefore be in reality at least as far out as the free-throw line. It could be beyond it, if it's above the floor, but it certainly cannot be between the basket and the free-throw line. This crucial piece of video detail shows that Fisher does not touch the ball until the second of the three live video frames above. Below are successive frames from the live video, starting from the one where Fisher first touches the ball, and running until he releases it.

Frame 1:

Frame 2:

Frame 3:

Frame 4:

Frame 5:

Frame 6:

Frame 7:

Frame 8:

Frame 9:

Frame 10:

Frame 11:

Frame 12:

Frame 13:

Frame 14:

Frame 15:

In my opinion, Frame 14 (which shows the clock switching from 0.1 to 0.0) shows Fisher apparently having released the ball—about as apparently as he touches it in Frame 1. If we take those two frames as the endpoints of Fisher's possession of the ball, then he has the ball for 14 minus 1, or 13 frames in all. At 25 frames per second, that works out to 13/25 = 0.52 seconds. The method I've used to produce our composite review I estimate to have an error of a frame or two in either direction, which works out to plus or minus 0.06 seconds; add another 0.02 seconds for the video re-encoding at 25 frames per second.

In addition, I should account for my status as a Lakers fan. (Who else would go through this much trouble for Fisher's shot?) I remember sitting in bed, having twisted my ankle in my own basketball game earlier that afternoon, and feeling pretty good about the Lakers until the fourth quarter, then anxious, then frustrated, then angry, and finally elated. It is sensible, to account for this possible systematic bias, to add a frame's worth of time to the figure above to yield 0.56 seconds. Note that the ball has clearly left Fisher's hands in Frame 15, which also shows the red light on the backboard going on for the first time.

To summarize, this video shows that Fisher had possession of the ball for about 0.5 to 0.6 seconds. One corollary of this finding is that the referees started the clock approximately 0.1 to 0.2 seconds after he caught the ball. This is entirely typical and in line with usual execution times; it would be unreasonable to claim the clock was started "late." It's certainly shorter than the three-quarters of a second that Garza claimed was necessarily human reaction time; after all, Fisher executed his entire possession in less time than that.

Final Thoughts

Some Lakers fans pointed out, in the aftermath of the series, that prior to Fisher's game-winner, Duncan's shot swished through the hoop with considerably more than 0.4 seconds left on the clock. See, for instance, this video frame:

If so, claimed Lakers fans, the Lakers should have had more time on the clock, possibly rendering the above dispute moot. NBA rules stipulate that the clock should be stopped at the moment the ball exits the bottom of the basket, including the nylon, not when it enters the basket. The frame above shows the ball exiting the basket with the clock switching from 0.8 to 0.7 seconds. By that token, it must have taken somewhere between 0.3 and 0.4 seconds for the referees to whistle the clock stopped after the shot swished through. Why did it take longer for the clock to stop after Duncan's shot than it did for it to start after Fisher's contact?

It's impossible to state for certain, but one possibility is that because it's less predictable that Duncan's shot will exit the bottom of the basket than it is that Fisher will touch the ball, the referees had to wait longer to be sure that the basket was made before whistling the clock stopped. Then, too, it's a whistle blow that stops the clock, as opposed to a button press that starts it again, and those two actions may well have different execution times. But it seems plausible that had perfect timekeeping prevailed in the final seconds, Fisher's shot would have been good by about 0.1 seconds.


  1. Very interesting. The smoking gun would be a photo of Fisher still touching the ball as the backboard lights turn on. No such photo is in evidence. Your claim, though, was that the clock started late. That's certainly possible, but I think you've introduced an unreasonable standard, a different standard from the one by which basketball is customarily played. For example, your reasoning about reaction time, and how long it takes a ref to start the clock when the ball is inbounded, is surely correct, but that's true of every inbounding event. Virtually every game features at least one or two instances of a basket just barely beating the 24-second clock or the end of quarter buzzer. If a buzzer-beating shot was launched with 0.2 seconds left on the 24-second clock or at the end of a quarter, and we took in account the fact that the ref started the clock on the most recent inbounds pass .3 seconds late, then those shots do not in fact beat the buzzer. That would be true even if the previous inbounds pass took place several plays ago. If the ball was inbounded at, say, 2:00, and remained continuously in play until a buzzer-beating shot at the end of the quarter, then the ref's delay on the inbounds pass two minutes previously is still relevant. In fact, the errors are cumulative, so if the ref's reaction is 0.3 seconds slow each time, and if there are 40 instances when his reaction time is a factor, then you'd be able to argue that when the final buzzer sounds, there are actually 12 seconds still to be played.
    However, I feel that, "as the game is customarily played," there is a certain quantity of human error involved, which would be unreasonable to "factor out." It seems unreasonable to introduce a "correction" to the ref's actions on this particular play, if we're not going to do the same on all plays. "As the game is customarily played," Fisher's shot was ruled fair because it left his hands before the buzzer sounded and the backboard lights turned on. It is possible the clock started slightly late due to a ref's reaction time, which is after all only human, but it's equally possible that the clock should have had more time on it due to earlier errors (as you point out in the discussion of Duncan's last shot--except that the discussion shouldn't end there, but include all possible instances when a ref's reaction time may have added to or removed time from the clock). As it would be impossible and unreasonable to analyze every such instance, so too it would be unreasonable to apply such standards to this one particular shot.

  2. I've always wondered why the referees didn't review the clock after Duncan's shot. I remember everyone at the party screaming at the TV that there should have been 0.7 or 0.8 seconds on the clock at the start of the final possession. To me, counting Fisher's shot as good means that justice was served.

  3. @Rob: Actually, I entirely agree with you; I feel I've probably incompletely conveyed my intent with this post. My main point was not that the clock started late, but that it did not start TOO late: The delay between Fisher's touching the ball and the start of the clock was entirely typical. Demonstrating that was the point of timing the sequence to that precise a standard--a standard that would be eminently unreasonable to apply as a matter of course.

    The discussion of the clock running after Duncan's shot was purely rhetorical--to further emphasize that the question of whether the entire sequence was completed within 0.4 seconds was moot, because by the same timing standards that some Spurs fans insisted on applying to Fisher's shot, he should have had more time anyway. Your point about that being true of all previous possessions is quite correct, but is unnecessary from a rhetorical standpoint.

  4. @Rob: Incidentally, in your hypothetical in which there are 40 clock events (starting or stopping) in a quarter, with typically 0.3 seconds of delay on each event, note that the delays on stops partially cancel out delays on starts (or vice versa). If they were each 0.3 seconds on average, then the mean effect would be nil.

    Whether that's actually the case is another speculation entirely. My intuition is that delays on stops are probably longer than delays on starts (because starts are more predictable in my opinion), in which case there would typically be ever so slightly less than 12 minutes of action in a quarter. This is not really consequential in a practical sense. (Of course, neither was this post!)

  5. ZAPRUDER FILM!!!! The first commenter even uses the expression "smoking gun"....

    Let me clarify that I think this is wonderful analysis. Certainly this moment in Lakers history looms large enough for me to want it analyzed to this degree of detail and precision. All things considered, I think a clock starting only 0.1 second late is pretty extraordinary reflexes on the part of whomever tripped the clock.

    Thanks for doing this, it has made my whole day better.

  6. Thanks for doing this. I've wanted to do this for years but couldn't find the video.

    I think the real lesson is that the game should just be played at a level of precision that refs can be expected to maintain. The clock should only keep track of seconds, not tenths of a second. In football the first down marker shouldn't be measured to an inch, just a half yard line or so.

  7. @Paul: Thanks! This analysis had its start in a comment on another blog (it might have been Mark Cuban's), wishing that someone would do a blow-by-blow analysis of the 0.4 shot. And here we are.

  8. @Steve: I disagree mildly, in both cases. I'm reading a book by Jimena Canales, entitled--coincidentally enough--Tenth of a Second, which argues that that interval of time is the shortest that people *can* properly apprehend.

    You can argue that maybe it should be two tenths of a second, rather than one tenth, but a whole second seems way too long. Suppose the clock reads 0:01, back in the day when the game clock didn't read tenths of a second. The true time remaining in the game (which is recorded in the clock hardware) could be anything less than or equal to exactly one second. If you knew whether it was really 0.9 seconds or 0.5 seconds or 0.2 seconds, that could dramatically affect your tactics. It wouldn't matter, though, if it was 0.16 seconds or 0.09 seconds or 0.05 seconds, which is why hundredths of a second is too short.

    Regarding football, I think of the placement of the ball at the new line of scrimmage as being essentially drawn from a probability distribution centered at the "true" end of the previous play. The impact of this is not felt until a measurement is made--that is, when the first-down markers are brought out by the referee.

    The upshot of this curious mixture of precision and inaccuracy is that the further you truly advance the ball on offense, the better your chance of making a first down--which I'd argue is exactly what you want as the rules committee. In most cases, the probability of a first down is either 0 or 1, but there's about a one-yard interval where the probability varies continuously between those two extremes. Quantizing the first-down markers wouldn't really improve that situation; it would only make it a staircase function. And you'd still have the same judgement call: If you go out at the 5-1/4-yard line, does the ball get placed at the 5-yard line, or the 5-1/2-yard line?