More videos with Professor Goldberg: bit.ly/LisaGoldberg

@shaggythehybridinglolingog9971

6 жыл бұрын

Is there a study on umpires / referees being biased? such as the chances of them being biased? or statistics?

@ehTrotcoD

6 жыл бұрын

Neil Abella don't have the link at the moment at this maybe isn't the bias you are thinking of, but Fivethirtyeight had an article about a study that showed umpires being more likely to make calls in extra innings that would bring the game closer to ending. So essentially a bias towards wanting to go home.

@maxyeung2417

6 жыл бұрын

I wonder what the critical region for proving the converse, a "frozen hand", is. Also, why is the simulated results bimodal?? What's the number of simulations?

@GianAgassi

6 жыл бұрын

What if we would look at this considering the start of the game: the more strikes you do in the first part (n shots) of the game (vs your average strikes %), the more likely you end up with an unusually high strikes %. (Biased?) experience tells us that when a player starts well, he usually also tries bold shots, that result in some zeros, but also grows in confidence due to relatively difficult attempts hitting the target and the end result is a great scoring % vs the average performance [this would be the hypothesis, in very qualitative terms, to be tested of course]

They should get a thermal camera and record the players' hand temperatures while they play.

@numberphile

6 жыл бұрын

ha ha

@ais4185

6 жыл бұрын

That's only on Sixty Symbols, get outta here!

@Triantalex

9 ай бұрын

??

My problem with this is if you did permutations on someone making 9/12 shots you could argue their sequence wasn't 8 or 9 in a row so therefore not on the high end of the "hot hand spectrum". But they did have a hot hand because they were shooting a higher percentage that game than normal. I think the study is incorrectly defining what we mean by hot hand.

@matteogauthier7750

6 жыл бұрын

I thought the same thing. Maybe should they check if players have more largely-above-average games than randomness can explain?

@Maharani1991

6 жыл бұрын

+

@bambam0802

6 жыл бұрын

This

@TheCavemonk

6 жыл бұрын

+

@alexjh12345

6 жыл бұрын

*

i love it when people dispel the false notion of hot handedness, that being said, shouldn't they be comparing "hot" games the a random sequence given their season FG%, not the game's FG%? If a player makes 25/26 shots in a game, the only sequence that would be considered "hot is if he missed the first shot, then hit every shot afterward, when a colloquial definition of "hotness" would say the player was obviously hot because they hit every shot

@physicaquanta6247

5 жыл бұрын

Yep! The study has at least three fatal flaws.

@matthewvaughan8192

4 жыл бұрын

If you take the stats game-by-game there are a tonne of variables in the quality of each team, so the results wouldn’t tell you much unless those teams played each other 10 times per dataset. So no, not really

@travisashby7780

2 жыл бұрын

It’s true

@sergiorenatoreyes6967

Жыл бұрын

@@matthewvaughan8192 so in a 7 game playoff series it can definitely be tested?

@matthewvaughan8192

Жыл бұрын

@@sergiorenatoreyes6967 Definitely not over 7 between 2 teams. The only way you could truly test it would be by completely changing the format of the Playoffs into a league system, so rather than a versus series teams randomly play each other and are given points for wins (sort of like the 2024-25 Champions League format.) Otherwise it’s pretty much always gonna have some obvious limitations

It's great that the Professor emphasized that you bring your own rules to an experiment and so you get an answer based only on these rules. Changing the rules may change the result so it all comes down to the compromise of what initial conditions we choose.

@Triantalex

9 ай бұрын

??

I think how you define "hot hands" is really important in a study like this.

@jessicastrat9376

4 жыл бұрын

I don’t see how it can be misconstrued

@vipertube7182

2 жыл бұрын

It can be

@vipertube7182

2 жыл бұрын

@@jessicastrat9376 this is mathematically accurate but it doesn’t tell the full picture well yes they will There’s a psychological aspect to it people tend to perform better because they have already been performing well not because mathematically they have a higher percentage when they have been performing well

@Triantalex

9 ай бұрын

??

I think this study faces yet another flaw. I'll use the game at 16:05 as an example. You are comparing the game with all the possible ways to score 31 out of 44 throws. But had Klay Thompson not have a hot hand, he wouldn't have scored 31 (maybe 25 or 28). So by doing the study this way, you are considering his point average as normal. I would suggest comparing his usual chance of scoring in all games with his chance of scoring during his streak.

@Lightning_Lance

6 жыл бұрын

Yeah, I agree.

@MsBombastik

5 жыл бұрын

You missing the point, yes he was lucky to make some shots. But did he became better shooter after he made these shots? And the statistic says no.

@Quantris

4 жыл бұрын

​@@MsBombastik "did he became better shooter after he made these shots?" how is that equivalent to asking if he was "in the zone"??? This seems like a weird strawman

Not sure if the rendering/KZread has had some part in it - but this video seems quite washed out - no contrast!

@Adlore

6 жыл бұрын

Jordan Colbert I believe it's raw footage which has not been colour corrected to look realistic. (Our eyes see certain colours more vibrantly, and this footage has not been corrected to account for that)

@therealDannyVasquez

6 жыл бұрын

Yeah. He's shot this with a flat color profile and didn't bother to correct it in post.

@tommihommi1

6 жыл бұрын

yep, looks like ungraded footage

@ze_rubenator

6 жыл бұрын

"has not been colour corrected to look realistic" In most movies and shows they amp the contrast up to a ridiculous amount and tint it in all manner of ways that are in no way realistic. You think it's realistic because that's what you're used to, but there's no such extreme contrast in the real world. Footage like in this video is, if anything, more realistic than most other processed video.

@tommihommi1

6 жыл бұрын

Ze Rubenator if this video looks like real life to you, your eyes are broken. Video is recorded with a flat profile, this enables large dynamic range. In post, the shadows and highlights are improved and a LUT is applied to give it the look the editor wants.

What about just hotness as increased shooting percentage and not streaks? Was his shooting percentage for the game a random fluctuation or was he really shooting better? Klay Thompson's field goal percentage that season is about 47%. Using a binomial distribution with p=.47, we find the probability of clay shooting that well to be 0.0005, or 1/2000. He has not played that many games. One should look at all games and see if the distribution matches what would be expected from a binomial distribution.

@lucasfonseca7078

2 жыл бұрын

Touché. I couldn't agree more with you

Another flaw in this study, aside from some of the others stated, is that it focuses on the top team and the top shooters in the league that already hit a disproportionate number of shots. If JR Smith all of a sudden hit 11/14 threes in a game, you can't deny he had a hot hand that night. This study completely eliminated phychology from the equation and treats a shot as a random independent occurrence devoid of a human element.

@wasdwasdedsf

5 жыл бұрын

uh... yes you can deny that... its a 14 sample size...

@angelmendez-rivera351

5 жыл бұрын

Matthew Jund I don't think you understood the implementation of the study. And 14 isn't many shots, so that's actually not even worth considering in a study due to the sample size being so small.

One thing that they fail to mention is that when someone has a "hot hand" they'll often times execute a "heat check", taking a far more difficult attempt than they normally would. This would surely skew the data.

Wouldn't it be easier to see hot hands in a bad player than a good player?

i think the meaning of being hot is a high FG%, instead of the actual streak of made shots. For example, if a player makes 21/30, we'd say he was scorching hot that night, even if the made shots werent one after the other.

Why are you treating a basketball shot as if it were a random event and not an application of the player's skill?

@derekmcdaniel6029

6 жыл бұрын

It's both. This is actually very common in statistics. A random event is simply outcomes where different results have different frequencies. A player's skill, roughly speaking will determine their percentage, but the outcome of a particular shot is still uncertain.

@angelmendez-rivera351

5 жыл бұрын

Because it doesn't matter how much skill you have, you CANT force a basketball shot into hitting every time. It's impossible. Because the physics behind a ball actually entering a basket are chaotic dynamics far more advanced than any amount of training could have you simulate. Not to mention that skill is by far not the only factor in any basketball game, let alone a shot. Players don't perform perfectly, even relative to their own skill, because of health, pressure, etc.

@JordanEMAW

5 жыл бұрын

The player's skill prescribes the percentages on the random event.

@philliberatore4265

5 жыл бұрын

@@JordanEMAW The player's skill is not necessarily consistent though the game.

@matthewvaughan8192

4 жыл бұрын

Phil Liberatore That doesn’t make his comment wrong. His is true and yours are true

I really like Professor Goldberg--it's good to see her again.

@onkilo

6 жыл бұрын

Came here to say the same. 😀

So, if you take 20 shots and hit all shots, then after random scrambling there should be 0 permutation from the average. Aka, not a hot hand. Should you not compare this string of 1s or makes against "the average" instead of against itself?

That guy near the end had 3 streaks of 4 in a row and 1 streak of 6 in a row. I don’t watch basketball, but that sounds like a hot hand to me. I think there’s some misinterpretation in what a hot hand is

TRIAL DESIGN ERROR!!! What would happen if someone makes all his shots and goes 20/20? Null hypothesis and a "streaker" would be the same, wouldn't it? By all human definitions, going 20 for 20 is REALLY hot! 19/20? 18/20? 17/20?

If I understood right, a player hitting all of his shots in a game won't have a "hot hand" by this calculation. that is obviously a bad definition of "hot hand". Same if the player only misses one shot. Missing the first shot would be a hot hand and missing one in the middle of the game would not, by this calculation. But obviously the player is hot regardless. C'mon if scoring 60pts is not being hot then what is?

The scrambling test literally made me drop my jaw in awe over how genius that concept is.

I wonder if the odds you take the shot go up while the odds you make the shot stays the same.

@CodyKing

6 жыл бұрын

Probability of making the shot isn't constant. There are lots of factors that going into the probability that a shot will be made. It is not just random like with gambling.

I love when math is given real world representation, makes it easier to understand for those of us (me) that were never too hot with math!

@TheTariqibnziyad

6 жыл бұрын

Science with Katie i see what you did there.

@muazkashif8554

6 жыл бұрын

Your method of gaining more subscribers (although distinctive) is proving to be quite effective. I Iike your videos, but do feel you need to lower the speed of your speech.

@tomsmith4542

6 жыл бұрын

typical victim mentality

@georgemargaris

6 жыл бұрын

math has the hottest curves!

@Pete-Prolly

6 жыл бұрын

Science with Katie you're not alone; I am cuz I prefer Math with little or no "known" applications LOL. Too often Math-majors try to justify it by saying 🤔"well, we might need it for 'this,'" instead of saying "I don't care if my field has 'real-world applications;'🤷🏻‍♂️ it's delicious without any."

The problem is that streaks of hot and cold shooting are not broken but one miss or hit.

15:55 I died a little inside when she said they didn't distinguish between foul shots, 2 pointers, and 3 pointers; different probabilities and changing proportionality. Nevertheless, one player scoring ~60+ points a game is a less effective way of winning; teams occasionally funnel the ball to a distinguished player on a seemingly good night out of player respect, fanfare, and adrenaline.

Brady, you're really on a streak with these great videos!

typical viewer: “They have all this data and methodology, but my FEELINGS though.”

This phenomenon was touched on in Gilovich's excellent "How we know what isn't so". A categorically recommended read for anyone who wants to understand how and why our brains make mistakes.

I think this is not an arena for solely interpreting events using math. He might miss a shot but it is because a guy who has a hot hand will be guarded more fiercely too. This creates more space and more likeliness of finding an easy assist. (The hot hand is not solely about making buckets. It is about creating space.) John Starks in the 94 Finals is the perfect counter example of the hot hand- the cold hand. His poor shooting basically suffocated the lane. And, he continued to choke for the remainder of the finals.

I understand the Gambler's Fallacy because it is mostly about luck when gambling, but basketball is a game of skill. By just looking at the data you are ignoring the skill and psychological aspects involved.

@mkyt2601

5 жыл бұрын

"Psychological" is so key. Sports psychology is becoming bigger and bigger. I appreciate the maths channel focus on just maths obviously, but confidence is a HUGE part of a player's capabilities.

@realitant

2 жыл бұрын

But if that confidence doesn't actually translate to an increase in shot percentage, does it really matter?

Seeing NBA Jam appear in a Numberphile video has to be the greatest moment of the year.

This video is soooo good. Thanks numberphile!

10:20 So this effect is about grouping results together and combining the relative results instead of the absolute ones, which would be 4 heads and 4 tails with an average of 0.5. I guess for this to happen it is necessary that the grouping is not random (higher chance of more than one H in one group). Very similar to gerrymandaring.

This is a great video, you should make more videos were people explain their own experiments/papers and the stories behind them

Can the permutation test actually disprove the 'hot hand' though? If the additional 1's in a streak were caused by a hot hand, then the permutation sample will include those additional 1's, so should be biased towards having more ways in which that could have happened through random chance.

One of my favourite Numberphile videos. That being said, I think in basketball looking for a hot hand shouldn't just be observed within a game, but it might also be very informative if the same checks for single players in each game were done for many players, but across a whole season. I think by looking for streakiness not just compared to the rest of the same game (which probably has a Hot hand being carried from earlier streaks), but should be compared to an average game for that player in some way. Or maybe across a season we could compare the number of full games where a player is hot, using a permuted season divided into random games as the Null Hypothesis / baseline. As a basketball fan I feel llike this bias would yield a much different story.

wait. but that's not how you accurately interpret data. shouldn't the last 2 be counted twice, since they had 2 times more data than the others. and that would make it exactly 50%

@cordlefhrichter1520

6 жыл бұрын

That's what I was wondering.

@francomiranda706

6 жыл бұрын

no its not. Youre finding the average of hits to misses after a hit in each sequence. then, youre averaging the averages. the point is that you are narrowing your scope to just the small sample size of one permutation, and then making an analysis of the average misses to hit ratios, but then comparing that (or averaging) to the total number of permutations. by doing this, you can get a sense of what is the "norm" in a set of permutations, and how representative each permutation is or is not of that norm.

@S3maxime16

6 жыл бұрын

Franco Miranda you can't just take a straight average of the hits to attemps ratios of every permutation to calculate the norm since some permutations have more credibility (more bernouilli trials) than others. You have to take a weighted average. It is the correct way to do it and it doesn't lead to skewed results.

@menachemsalomon

6 жыл бұрын

I think you could, and part of the problem the Prof was describing was different ways to analyze the same results.

@coopergates9680

4 жыл бұрын

I agree, 4 heads and 4 tails were written down. The original question was what happened after each head flip, not the fraction of heads flipped after a head flip in a sequence of consecutive flips. To word it another way, what is the average number of heads flipped after a head in any possible sequence of three flips? There are eight instances where a head is followed by another flip, and four of those cases result in heads. Not only is the chance of a head being followed by a head 50%, but flipping a coin three times finds four cases of a head after a head among the 8 possibilities. Three flips in a row will produce half a head after a head, on average (which is unrelated to the "streaking" notion).

So timing ie fast actions could be interesting. I also think what happens post a missed shot is important. How bad was the miss? Rebound and score (assist) vs other team scoring.

HHT and HHH where counted as one measurement, even though they have 2 outcomes each. If you count them according to the number of outcomes, 2 each, then it is 4/8 = 50%

@cordlefhrichter1520

6 жыл бұрын

I was wondering about that too.

@PaulPaulPaulson

6 жыл бұрын

Gerrymandaring

@Edouard16

6 жыл бұрын

Yes, I’m shocked that they could make the mistake of averaging statistics in subsets with different occurrence probabilities. I don’t know if their statement that small finite sets have a bias for change is true or not, but the way they try to prove it is broken. What happens after only takes into account, by definition, the next event, not the chain of three events.

@Flati36

6 жыл бұрын

Yes, this bugs me to no end. Shouldn't they do a weighted average?

@S3maxime16

6 жыл бұрын

Came to the comments section looking for this. Anyone with a basic probability understanding should not make that mistake.

In the coin flipping example, why is it interpreted as 6 measured groups of 3 coin flips rather than the actual 8 measured individual flips? When looked at as individual events, there are 4 of each for even odds as expected.

@ultimate0levels

6 жыл бұрын

i think the professor made a mistake.

No basketball shot ever has 50/50 chance of going in.

I find that videos on sports statistics tend to be hit or miss.

I'm an engineer... I appreciate science, and I don't usually use ''feelings'' as an indicator of anything ! But ... When you play ball, sometimes you just feel it !! The rim just seems bigger !! Something in your mind when you kind of achieve a level of focus, the game slows down !!! Yea I know I'm not disproving anything here but anyone who played ball knows the feeling !! Numbers don't lie, but they often don't tell the whole story either !!! Great video ! I wish numberphiles would do more '' sports stuff'' !!! Cheers !

@Chasky33

6 жыл бұрын

I Had the exact same felling! Another engineer that play ball.

very interesting. i’d love to see more studies on this, although last years Warriors are probably the best case study available. From a viewing perspective, that was the hottest shooting team in the history of the sport. The only other team that would be close would have been the 2015-2016 Warriors.

I'd be interested in seeing a sequence of 1&0s that would be considered a hot hand

basketball is just coin flipping - you either make it o r you don't. There is no skill or strategy involved. That's why it's perfect for statistical studies.

@Bill_Woo

6 жыл бұрын

I see what you did there :)

@pvanukoff

6 жыл бұрын

lol -- exactly. why even play the game at all? just setup a series of coin flips. so much more efficient.

This was a great study that I appreciated reading when it came out overall and love seeing her on the channel! My only "issue" is that the study doesn't seem to take time and frequency of shot into effect which are large parts of the "hot hand" which is why I would guess many coaches and players have problems with these studies. You're not taking into account if these are shot in succession or with a small vs large number of possessions in between. So this data looks at 2 shots in 2 successive possessions to be the same as a buzzer beater halftime 3 being shot "next to" their next shot being taken a few minutes into the 3rd quarter and the same as 1 shot being made then the player not shooting for the next 6 possessions (for example) and then making another which due to time and number of possessions in between wouldn't be considered a hot hand generally. Also, no one would consider this 2nd option a hot hand but the study could consider it to be one, while the 1st option could be considered the beginning of a hot hand by any natural observer/logical argument.

More of a psychological problem than a mathematical one. "In the zone" literally means having the most optimal mental state for the given kinesthetic activity. Whether that translates into "luck" or not, now that's the mathematical part of the problem.

one trouble I have with this is the mathematicians are assuming scoring a goal or basket in sports is a purely random event. It is possible that on that day in those conditions the variables have aligned to allow one particular player a better than average score rate.

I don't know why but I love the term Null Hypothesis. It's music to my ears for whatever reason

"Clay has made 40 points so far, but actually he's having a cold hand, let's stop giving him the ball"

Coin flipping is different from a competitive endeavor like a sport. People playing a sport or other competition can find themselves "in the zone." It is a mental state. A confident state of mind. A self-hypnosis of telling your unconscious that we will do well today. There are no thoughts about other than the current game. No distractions. No relationship problems thought about. No money worries come to mind. No teammate problems. Feeling good. There are days when players are fully in the game; these are days when players are above their average. Days, one might say, that they had a hot hand. If a player normally makes 60% of their shots then making 40% is a not a hot-hand day. If a player normally makes 30% of their shots (they're on the team for defense) but today makes 40% -- bingo! a hot-hand that day. Depends on what is meant by "to have a hot hand." If a player is having a lucky streak -- not a skill streak -- shouldn't we avoid giving the ball to him? Skill is so much more dependable than luck. Say a player makes 6 shots out of 10 attempts. When scrambled (1001110011) there a fixed number of each length of streak. In the example one of each length 1, 2 and 3. If a player scores 7 out of 10 the average streak length must be longer. A "streak" -- a "hot hand" -- may well indicate higher than ordinary skill, and not just luck.

I love how she doesn’t make any rash claims about her research and simply presents it as evidence. THIS is how everyone should do science.

@matthewvaughan8192

4 жыл бұрын

This is how everyone does science. Not sure where you get your scientific information from but it should be considered an anomaly when a scientist does anything but

Another thing to consider is that it is easy to remember the successes than the failures, especially if a failure quickly follows a success, or something dramatic happens after the miss, such as the opponents moving the ball to the other side of the court. As such, that could skew people's perception of a scoring streak, making them appear longer than they are. It's like how you can read a book for what feels like 5 minutes, but ends up being several hours.

I love that they keep using the 1967 finals footage as the video chapter title background. Great series

While many might be prone to believe a streak represents an underlying tendency more than it does, there is plenty of reason to believe that humans undergo periods of higher performance. We have objective good and bad days, we undergo constant neural plasticity, so I would say careful not to be too eager to flatter yourselves that peaks don't happen just because of short interval constancy.

This test (the permutation test) assumes the ratio between 1s and 0s is always the same. However, the idea of the "hot hand" is that it increases this ratio, isn't that true? how can you explain that and how you can test for the change in this ratio?

I would think that moving averages of some kind need to be involved. Missing a single shot in the middle of a streak shouldn't be so penalized when it comes to quantifying "hotness" (at least as I understand the conventional definition of it)

The 3-flips sample not only has an arbitrary stop after 3 flips, but it also has a biased "start" (as the first flip is only used to see if it is a H or T and not as a scored result). If the experiment was adapted to be cyclic (e.g. if flip 3 = H then write the result of flip 1) then will the bias not then be removed?

Not a sports fan, so I don't know this 'hot hand' mythology... But I doubt that this mathematical permutation test has anything to do with testing what most people's intuition is about. The average spectator, I would think, is saying, man, this guy, Thompson is playing great tonight (in comparison to his usual performance). And, lo and behold, the spectator is right: after scoring a higher percentage of his shots in his opening than usual, he continued scoring well throughout- ending up scoring sixty points in a single game. I would be surprised if the average spectator is saying that because the last three shots were baskets, that the very next one will also be one, but rather, that some nights he scores a smaller percentage, other nights a higher percent. And this seems to be one of those nights that he is playing better.

I appreciated her explanation at the end that the hot hand can't necessarily be explained by the parameters of her study. I've watched Klay's 60 point game a few times, and while he made a few tough shots that would normally be attributed to having the hot hand but could be explained by randomness, what I found most striking about his performance that game was his uncanny ability to get open. It was like he was thinking the game two steps ahead of everyone else. I've never seen him - including the 37 point quarter! - play a game so effortlessly. His makes vs misses weren't what defined his hot hand that game for me, it was everything surrounding the shots.

Surely you could use their fg% and see whether they shot above or below after a streak of shots and misses

You seem to assume the chance to hit remains the same for each shot, but other things such as skill, morale, general well being also plays a huge role in the chance a player can make a shot, its pretty much impossible to calculate that.

@lydianlights

6 жыл бұрын

Actually the permutation test they talked about addresses that problem pretty well. By shuffling the data and then seeing how the actual data compares to the scrambled data, you can tell how close the original data is to random, independent events. The shuffled data has everything in common with the original data except for the order, so that way you know that you have isolated exactly what you're trying to test for.

@Bill_Woo

6 жыл бұрын

No, Lydian, his proposition is valid. Your fundamental fallacy is that you dismiss the presence of fallacies :) Like most smug statisticians (not that YOU are), you profess that the selection set (or sets) reflect reality. And that what is true "in general" for the whole (or on average) is true for subsets. Not always. And not here. BTW, I R a statistician - a high level one, not some hillbilly trolling empty pontifications. Further, statisticians robotically (and quite unintelligently) issue conclusive pronouncements when it all rests on *assumptions* of underlying probability. You can shuffle data sets until cows dance the macarena, but you can't pinpoint the probability of an athlete hitting a shot. Even if you measure 80% of all shots (and I state to you, with great certainty, that the analysis omitted over 99.99995% of them). Like typical (if not the overwhelming majority) statistical smug pronouncements, the cards all collapse if the assumption does not hold. I disdain smug pontificatory statisticians who profess nonsense under the cover of [ *inapplicable* ] statistical principles. Not that _you_ are one of those, Lydian :)

I understand what they are trying to say, but there are so many other factors that aren't accounted for in this experiment when it comes sports that it's impossible to be accurate.

@Merahki3863

6 жыл бұрын

JESSEverything in the context of the hothand the only variable assumed is a streak. No other conditions have been associated with it so the rest of the game doesnt matter in analysis.

@pkacc1

6 жыл бұрын

The assumption is that basketball shots are like coin flips - random, independent trials. They are not.

@lydianlights

6 жыл бұрын

@Xibalba No, the NULL HYPOTHESIS is that shots are random, independent trials. The data is then analyzed to see if it is consistent with this null hypothesis. As it turns out (by the measures of this study at least) it seems to be. What I'm saying is that it is not an assumption, it is literally the thing they are testing.

@Unimatrix69

6 жыл бұрын

And that's a false assumption. The shooter is not playing by himself - the other variables that constitute a game situation CANNOT be ignored.

@tedlemoine5587

6 жыл бұрын

JESSEverything you are correct the human element and the emotions of confidence or self-doubt cannot be factored in mathematically. I'm a mathematician and former athlete who can see the difference in real-life vs a formula

Wouldn't you have to take the data set as a player's entire shot history in a season? If Klay Thompson shoots 31 for 44 in one game, of course the observed shots will not differ from the null hypothesis, because even the null hypothesis would be considered a "hot hand" in that his shooting percentage was very high, so he would naturally string together several shots. Only a player who normally shoots 70% can expect this turnover after each make. If you take these results in the broader context of the whole season, you might find different results. One game he could shoot 31 for 44 but another he could shoot 2 for 13, and these performances would be more stratified than a null hypothesis for the entire season. Additionally, the utility of these findings should have an impact on whether the rest of the team should pass the ball to the "hot hand." But regardless of whether Klay's shot history varies from his own null hypothesis, he likely would have a much hotter hand than compared to the team as a whole. Maybe a better metric would be to measure a player's full game percentage as a function of his first quarter percentage. That would indicate whether a player who is hot or cold early in a game is likely to repeat this performance, suggesting whether or not they should be taking shots.

This is a terrific video because it helps to explain what I’ve come to believe is the fallacy of the “hot hand” in sports. But more than that it alerts me to fact that small sample sizes may behave differently that big sample sizes, and that most problems are more complex that appreciated. Human aren’t likely good at understanding randomness, but we are good at picking out patterns. The problem is those patterns may in fact be misleading.

I would have thought that statistical significance of a hot hand would be less related to single strings of shots, and more about periods of statistically higher percentages, so something more like a standard distribution and deviations based on a player's usual shot percentage over a period of time. Aka, a prolonged period more than a standard deviation above their usual shot percentage would be considered a hot streak, and a standard deviation below would be a cold streak. Also, it's probably worth considering that from the standpoint of this study having a hot hand is not the same as being in the zone, and perhaps that is why it specifically looks at the significance of shot streaks instead of percentages or other stats to determine whether it's there. Because the stats do seem to show that some players have significantly different shooting percentages between a good and bad game which is why they are thought of as streaky in the first place. What this study says is more that the idea of being on fire with your shooting doesn't necessarily mean you will hit long streaks of shots and mostly seem to show that when a player is in good condition they hit a higher overall percentage, but the statistical significance of the actual streaks doesn't change much. Being in the zone is essentially the idea of being in the perfect mental state while playing, while the hot hand refers specifically to score streaks. Similar ideas, but not the same, and showing that there isn't a statistically higher rate of shot streaks doesn't account for the overall higher level play that someone who scores 60 points in a game has, and streak or not 60 points is definitely a statistical anomaly. I suppose that's all part of what makes statistical analysis of sports interesting though, because humans and their mental states tend very much not to fit neatly into data sets.

I LOVE me some probability theory! Thanks for this! I would like to see a similar discussion in combat sports, like boxing or mma, that demonstrates how much is luck vs skill in winning belts.

I'm a simple man. I see a Numberphile video I search the comments for Parker Square jokes and refrences.

Brady, I love your videos. As an avid basketball player and a mathematics graduate, I think these studies are not measuring what basketball players and coaches may be recognizing as having a hot hand. These measure temporal hotness. Something that comes and goes with any given shot. From experience, there is definitely such thing as having a hot hand. However it typically means that I am simply shooting with a much higher percentage than normal. If you are confined to a particular game, you are measuring within my “hot zone.” So I would expect general randomness. What I would like to see is a study of “hot games.” When a player has a hot or a cold game, is it just part of the general randomness you would expect from an X% shooter, or were they actually hot or cold that day? I suspect that outside factors such as sleep, injuries, diet, personal life... all play a roll on making a player extra hot or cold on any given day. When trying to implement a strategy around this, the trick is probably differentiating who is actually hot or cold vs who is simply experiencing a moment of randomness.

Fantastic analysis. Prof knows her stuff :D

Great video as always, but is the common definition of "in the zone" really just 100% accuracy? I have always considered it to be a boost in accuracy. If you use that definition and study the numbers, would you get different results? Obviously that would first require a measure of what a series of shots is considered. For instance, does Steph Curry normally shoot 3 shots within a 5 minute period or 6 shots? Does he shoot more when he thinks he is in the zone? Does his accuracy (shooting percentage in this case) increase from his average? Should a hot streak be capped by two misses in a row or >50% missed shots in a series of shots? If I feel like I'm in the zone, I don't lose the feeling with one miss; i assume that there was a variable that I didn't account for at the time.

Something that would go along with this is in the shooting sports. Lones Wigger a Smallbore shooter talked about the mental average. You would start out shooting really well but your mental average would cause you to start shooting poorly to bring your score back to your mental average.

Wouldn't looking at hits after misses (as opposed to hits after hits) give exactly the same result? ( 8:05 to 9:50)

Is it harder to measure whether really good (or bad) players have a hot hand? Do you need a much bigger sample size? For example someone who had a 100% shot rate would have a 0% chance of increasing their probability of scoring a basket. Seems like you would want to look at a distinctly average player scoring 50%. i.e. player with most 'randomess' in his scoring. For players scoring 70-80% average then it seems more likely that you would get negative hot handedness since otherwise their probabilities would tend to 1

Can this method be used to catch cheaters or people shaving points?

Some others have touched on this, but the issue I find with these studies is that they don’t define a hot hand in the way I think most basketball fans would. In these studies, a hot streak ends when a player misses a shot, whereas in reality there are plenty of situations in which a player can miss a shot but continue a hot hand after that. For example, a player might be on a streak, miss a shot, get a rebound, and score on that same possession. In this experiment, that player’s hot hand would’ve been deemed over, but most people watching would actually think that the player’s hand is ever hotter at that point! In this way, I think the term “streak” is misused by sports fans, when the term would more appropriately be “hot trend.”

"Splash brothers" on numberphile! Never thought I'd hear it. Fabulous start to my day.

Wonderful video. Absolutely fascinating.

I feel like we're missing something by not comparing the players scoring in each particular game to that player's average. If a player is suddenly making 80% of their shots when their overall average is closer to 50% of shots made, that's a huge difference. Additionally, the timing thing seems incredibly important to me. Are those streaks being broken after a break in the game? Are they being broken when the player hasn't had a chance to make a shot in a while for whatever reason? What are the conditions around the streaks being broken? At what point do the conditions to continue a streak no longer exist?

Doesn't the fact teams are specifically playing to and strategizing for this "hot hand" effect change the results? If a player keeps getting passes, or maybe getting situations set up for them, wouldn't that increase their chances of scoring? Would be interesting to see a comparison between players who were believed to have the "hot hand" versus players who were not and were just shooting without any of that setup.

Absolutely excellent, brilliant formulation, very accessible.

I love listening to smart scientists setting up their design. Like, as a future nuclear engineer hopeful, experiment design seems like 90% of the challenge.

With the given string of 0s and 1s, it is very easy to see that the algorithm (or whatever you wanna call it) would give us a bad rating. The string with the best rating is one starting with 0s and ending with 1s, without anything mixed in, while the worst string is the one where 1s and 0s are mixed together as evenly as possible. The given string already looks pretty even, only at the end it gets a bit strange.

I am a bit confused about the "Fraction of H after H" measure. I really wanted to at that point count the number of H's after H's and divide that by the total number of H's that had something after it. And that would be 4/8. Why do we care about the fraction within the 3 length HT strings? I started getting confused, because what we are asking at that point is confusing to think about: "In all possible HT strings of length three, what is the expected number of H's after H's in an arbitrary HT string of length three?" Though I think typing this may have helped me think about what we are doing. We are looking for the average percentage of shot made after shot (H after H) that occur in HT strings of length three that have at least one shot attempted after an shot. Its still kind of weird to think about though, because of the fact that certain strings are discarded, and that each string doesn't have the same amount of pairs that matter. Awesome video by the way. I really love thinking about my statistical thinking biases.

@hexalspace

6 жыл бұрын

Thinking in a simpler way, we want the average (HH tuples in string/HH tuples possible in that string) for HT strings of length three. However, I guess it's not obvious what to do with the 0/0 strings. I guess throwing them out makes sense since they have no data to offer what we are interested in, the HH tuples possible... Anyone have any nice ways to think about this?

@hexalspace

6 жыл бұрын

Also am thinking that as the length of the string goes up, the number of 0/0 strings you have becomes much less significant to the data set, as you will always have only two (all T's and all T's followed by one H). So even if I agonize over them, I guess they are not so important to focus on meaning wise.

With the permutation test, it doesn't take into account that the amount of successful shots could've been different. Think about it, if you were truly to have a hot hand, you would get more total successful shots. Although this does depend on the 'hot hand' being defined not in terms of just the previous X shots being successes, but rather, the idea that successes have a positive feedback loop in terms of overall performance. For instance, if you miss 10 times in a row, you'll feel really bad and discouraged, which will impact your mental state, and cause you to play worse. Whereas if you're doing really well, you feel like you're on top of the world, and you will perform better than average. After all, this is how the hot hand was stated near the start of the video, that you get a better performance than your lifetime(or seasonal) average when you're doing well. It's an important distinction to make, and I do think the "hot hand" as I just described it is a thing. Now back to the permutations test, this definition certainly doesn't allow for an unbiased permutation test. Because the permutation tests are based off a set amount of successes out of a total. So, in a situation where a player got the theoretical hot hand, they would perhaps make 30 out of 40 shots, whereas if they hadn't gone into the hot hand, they'd only make 20 out of 40. No permutation in the test would match the latter case, and it would simply conclude that "he didn't score most of them in a row, therefore he didn't have the hot hand". While my last paragraph was entire a repeat of an earlier sentence, I just hope I make it clear; it very much depends on what you call the hot hand.

Klay Thompson scored 37 points including 8/8 from 3, which was the most points and 3's ever scored in a single quarter in nba history. He also didn't even play the 4th quarter of that game. For reference his average is only just over 20 a game. When we say a basketball player has a hot hand it is also in reference to a short period of time within that game. It does not last a whole game. What you should be doing is looking at his most average game, take the string from that and compare that to his string from this game. I can guarantee you'll find the sets to be significantly different. On a side note seeing a numberphile video talking about the NBA literally made my day

I LOVE YOU on a more serious note, how well does the sequence comply with the idea that the guy just scores X% of the time?

I really think that you're looking at hot hands from the wrong angle. We need to collect information from people if a player was having a hot hand right after a game (without commentary). Then we can try to train an universal appropriator to predict "hot handedness".

I would have thought that at highest level the biggest influence on the outcome of a shot would be all of the regular training and practice, followed by all of the lifetime experience and then followed by any off the field or personal events affecting the player's overall mood on the day. Perhaps at an amateur level where there is less training and other experience the confidence and inherent experience of having scored several prior shots would have a big impact but it shouldn't be as significant for the top pros.

Like a famous German football player said, football is like chess, without rolling dice. I think the numbers are enough evidence to show the effect is minimal, but there are two psychological effects not canceling out: to fear the hot hand opponent on one side and getting confidence in repeated success on the other side. Also, everybody knows feelings of "being in a flow" in any type of work. So there is some intuition about this telling us this is not necessarily wrong. Overall how good a player or team plays is not just random anyway. Therefore I like this kind of precision about how to tackle the statistics to confirm or disprove within an objective context and questioning the objectivity. I like the outlook of how that could have another turn when taking into account the timing.

With a bit more data you can make this more accurate by adding an additional variable because timing is important. In fact, that's one of the main theories about a "hot hand". If someone makes 3 in a row, then you don't give them the ball for 5+ mins or it's half time, maybe they get "cold". Maybe a future study will find out a "hot hand" does exist, but it's limited within one or two minutes in between attempts while the muscle memory is still strong. The study might be something like, if a player makes 3x in a row, is there a statistical significant variance on the 4th shot based on the closeness in time to the third shot? Just a thought.

Choosing to shuffle the order of the results to me is stupidity. Such an operation assumes no causality, and discounts the psychology of making a shot. Time is not randomly sliced up, it proceeds from the past towards the future. When one makes a shot, it builds confidence and therefore influences future successes. So reording event history to do some sort of statistical analysis seems nonsensical.

The biggest problem I see with comparing coin flips with basketball shots is the amount of control the player has over the outcome. A great player can have a huge influence on whether his/her shot will land or not, while a coin flipper just has pure luck to count on (or something really really high, given the low amount of control one can have over a coin flip). You can't really compare the two. If you see a great player getting a lot of the shots in, you'll count on the possibility that their body and mind have reached a point while that game is going on, where the player seems to have such control over the shot, that it'll be more likely for the shot to go in. It's called building consistency. If you try to deny that such an effect can happen temporarily during a game, the only logic conclusion you can get after that is that training to build consistency doesn't matter. Because if you argue that one can't improve their consistency (even if temporarily) during a game, you're claiming that practice does not improve your consistency at all, because if it did, even if just a small amount, then the effect would happen during live games as well, which contraditcs your original assumption.

"They're dealing with small, finite samples - and Small stuff does not behave like the big stuff" The mathematical equivalent of the divide between relativity/quantum mechanics?

@Bill_Woo

6 жыл бұрын

Alas, you are correct, but others are in charge of the asylum!

This is a really super simplistic view on such a complex problem. I understand the need to make assumptions, but to somehow determine 'hot handedness' from a score of 1's and 0's from professional mathematicians only strengthens my view that what is a statistical problem is best solved by statisticians who take into account more variables and their co-dependencies. Mathematicians only seems to be interested in a basic prima faccia view on things...

The biggest variable these studies seem to ignore is the average defensive ability of the opposing team. So like a hot hand for shooting, a team could also have a streak of blocking shots, stealing balls, etc. Also, what type of shot is not considered, nor the difficulty of a given shot in the pros; for exampke, one can score the same number of points with fewer three point shots than two point ones.

Excellent video!

Has anyone ever studied this using shot quality as a factor? A lot of stats nerds in Hockey spend a lot of time talking about high or low danger shots and passing feeds.

Maybe someone can help me understand a certain aspect of this. I realize using a coin toss was said to be a "toy example" but yet they keep referring back to shots as random. Making a shot in basketball has absolutely nothing in common with tossing a coin. Skill and confidence, to name to differences

@matthewvaughan8192

4 жыл бұрын

As far as numerical datasets go, skill & confidence are completely random. Of course one player will have a higher chance % of scoring than another, but we’re comparing the data of individual players against themselves, so the difference in skill makes absolutely no difference. Confidence is essentially the grounds of the hypothesis. That’s what the data is testing: does confidence grant players more access to their skill and therefore make players more likely to score? Repeatedly we find that the answer might be a rather surprising no, at least in basketball

@kejtos5

2 жыл бұрын

@burtalism Yes, but the number of throws necessary to meaningfully increase skill of a basketball superstar is in order of magnitude higher, then the number of shots the player can make in a match.

I have no interest in sport but this video was captivating!