We analysed 4,678,387 NBA shots 

Nice video, thanks for inviting me, and thanks for spreading the word on SoME3! I was surprised that an NBA player would have to be 2 feet from the hoop to get the equivalent expected points as a typical three-pointer.

Matt Parker is now officially a baller

I feel like part of the reason the slope of the accuracy chart beyond the 3 point arc is so gentle, is because the only ones crazy enough to take the longer and longer shots regularly are the players like Steph who have the skills to back it up, and so the data is skewed/inflated

Grant highjacking the video at the end by asking intelligent questions.

As a primary school teacher, this is a huge help to answer the kids as to why math is important even though they only like sports :D Thanks Matt!

Matt literally gave Grant a Parker Jersey

Things I learned: 1) Sport math is fun. 2) Grant has some GUNS. Damn.

I sort of wished Jon Bois would be a part of this as well!

I love 3b1b's casual appearance in the beginning of the video. Anyone else would take a good chunk of time from the video to properly introduce another youtuber but Matt's just like "Hey Grant, wanna do some plots" lol

4:00 A heatmap would have been a better representation. You would visually get color variation of the hotspots as they move around over time. Trying to visualize it in 3D in a weird bar chart format makes it a lot more difficult to actually tell what is going on.

Let me get this straight… Grant is smart, handsome, has an awesome voice AND IS RIPPED??

Your accuracy graph is "distance in inches" instead of what I'm assuming is supposed to be feet. It's got me dying over here imagining you and Grant missing 80% of your shots from 20 inches away.

I can't wait for the uncut video on the second channel where Matt performs 4.678.387 shots. It's going to be a blast!

Let’s all take a moment to appreciate Grant’s sick burn on Matt at 18:10

A 3-point shooting percentage of 26.7% is statistically as efficient as 40% in the mid-range due the 3/2 factor increase in points scored.

a big thank you to Tim Chartier for stating that the questions makes the statistic, not necessarily the numbers. (which matt proved with his infamous stonehenge/woolworth (or whatever shop it was) video)

5:56 Nailed it. I can't be the only one who genuinely loves this cheeky posturing on a shot that goes wide.

I think Grant has a point. Prior to Warriors' run in 2015, three-point-shooting-centric offense was frowned upon because relying on at least one of your players to make > 35% of his threes, at a high enough volume, for almost 100 games, is ridiculous. Obviously then the Splash brothers came along, and advanced analytics (whose mass usage was at its early days at the time) supported the increasing usage of threes (and dunks), which I guess made threes cool even among big men.

Subtle parker square, great humor, killer lyrics. Keep it up!

You could make a heat map of accuracy of shots on the court to also see the effect of angle from the basket,

Oh hey this is my area of specialty! My official job title is "Lead Nerd", but basically I work doing statistical modeling (among other things) for an agency that represents a few players in the NBA. This particular problem is one that's well-known in our canon. The best way to represent it, IMO, is not as a direct expected value problem like this does, but rather to treat it as a diminishing marginal returns problem. Basically, the xth 3 point shot you select is more likely to go in than the x+1th. As a result, there is (well, technically this is "may be", since at sufficiently low enough expected values the curves might not cross -- that is not the case here) a point where you are better off selecting the 1st midrange shot, so you do end up with a Nash equilibrium in which all three ranges of shot are taken. Grant's question regarding increased usage (which is the term used for players taking more or fewer offensive touches) is actually also fairly well-studied. The easiest way to think about it is as an economics style trade-off between usage and points per shot, but that also becomes complicated by rubberbanding effects. It's fairly common for high volume players to record higher than expected points per shot because they're able to continue shooting after a poor start, where a lower usage player that misses his first 3 shots probably doesn't get to shoot again that day. While Tim's answer about sample size is important, it's also important to recognize that context effects are non-linear, non-monotone, and non-independent. Also, it's worth noting that the "location first" approach has been heavily criticized in recent years, because people (Kirk Goldsberry of ESPN chief among them) have argued that the homogeneity of distance is also creating homogeneity of play. However, the manner by which the ball gets to those locations is as diverse as it's ever been, so reducing it to just "oh the shots are coming from this exact distance" does a disservice to a game that we all love. As for the discussion on coaching and how to present numbers to decision-makers, that's a delightfully fun topic. I've been told stories by my peers in front offices of how they'll intentionally withhold specific information from coaches and players so that it doesn't get misinterpreted. For me a lot of the time I have to interact directly with players and families, and boy is the first time you have to sit down with a player and say "even though there are legitimate basketball reasons for this number to be low, you really do need to work on it" harrowing.

as a very non-sports person this was unexpectedly very interesting! would love to see the other-sports-mentioned versions of this video, particularly swimming!

39.6% somehow reminded me of the Standardnormal-distribution density at the center (1/sqrt(2pi))

Always excited for a Matt Parker chart party.

Matt Parker is channeling Jon Bois and I am HERE for it!

I'm very excited whenever Bec shows up. Grant is a special treat as well and I'll be equally excited for him as soon as the three of them start a podcast.

"You shoot once and you make it, you're amazing, you're 100%." This is why you'll often hear stats like, "Dominik Hasek is the all time leader in save percentage for players with at least 200 career NHL games," instead of just, "Dominik Hasek is the all time leader in save percentage." There could be someone who got called up from the minors, played one game on the current first place team against the current last place team, so he didn't face a whole lot of shots, and had a shutout. His save percentage would be 1.00, beating Dominik Hasek's 0.922.

Grant is ripped holy moly!

Other ideas: Did you consider running an analysis evaluating distance and angle? S.T.: y = distance*x1 + angle*x2. This would allow one to find the 'highest scoring zones' on the court. It would also reveal whether there's a 'left-right' bias, perhaps due to players' handedness.

Part of the expected value is also the rebound percentage. Making 40% of your 3s may be offset by recovering the ball less often than close shots, which lets your team try again. The current maths take the assumption that you get zero points on a missed shot (guaranteed turnover). I wonder if that is a valid assumption?

lol, one thing is in not mentioned in the video: The main reason why 3 pointers became so popular has to with a rule change: In 2004 „hand checking“ was banned. Afterwards as a defender you had be much more careful in order not to get a foul (see below). This made it easier for shooters to throw from outside. „A defender may not place and keep his hand on an opponent unless he is in the area near the basket with his back to the basket. A defender may momentarily touch an opponent with his hand anywhere on the court as long as it does not affect the opponent’s movement (speed, quickness, balance, rhythm)."

very interesting and pointed questions from Grant. I love how curious he is and how much he wants to get a better understanding of the topic. It felt like Prof. Tim was giving a thesis defence with Grant being the dept. head grueling him lol.

Just surprised they didn’t mention that the delay in increased shot distance coincided with the first kids who watched NBA games with a 3 point line finally making it to the NBA.

I recently read “Humble Pi” and loved it! Found it at Barnes and noble and instantly decided which book I wanted. :D

As a big fan of both basketball and math, it's nice to see you make a video on the NBA. Great video, as always! Morey ball!

As someone who uses data to provide insight and advice to clients for a living, there's a lot to be said for having a team of people who understand the data but not the context, people who understand the context but not the data, and people with a reasonable understanding of both. As someone from the first group (but steadily gaining the subject matter knowledge to move into the 3rd group), it's often frustrating when a client believes the data/insights are wrong because they do not conform to what they want to believe.

What I find most interesting about this is the "plateau" of accuracy against distance. I imagine that accuracy is a function of the probability of a shot going in *but* also the effect of defense. Because the probability of a shot going in decreases as a function of distance, that must mean that the effect of defense decreases as a function of distance. The decrease in defense as a function of distance makes all sorts of sense because in a "semicircle" like a court, defenders have to move farther distances when they are farther from the basket. What a cool video and data. Thank you.

Feel a little bit better about my 27% rate of draining threes in pickup basketball. Plus, when you play ones and twos, the mathematical leverage of making one is even higher! 2x as many points vs the 1.5x in the NBA.

Great video topic as always. You’re great at picking varying topics of interest

That was one of the most interesting videos I've seen in a while. Thanks!

It's nice Grant gives us so much time to prepare for SoME, since we aren't not even in winter yet!

I'd love sequels to this on 2 point conversions in american football as well as fourth down descisions in the same sport

Thank you, Matt and Tim, this answers a question I have often asked myself

This channel keeps answering questions I didn't know I had. Always interesting

Somewhat random, but I'm glad to see the subtitles get more reliable! The chat at the end and the song lyrics weren't great, but they're way better than they used to be - I know there were ideas for a new subtitle system on the channel going around last year, so this is nice!

Grant is absolutely yolked

The even more mathy (mathsy?) and updated version of the Chart Party video on this topic and I'm 100% here for it!

Note the "Parker" squared on the back of the jersey -- certainly a throw back to an earlier video with the "Parker Square." LOVE IT.

I love the contrast between Matt and Grant in the interview, Matt always has the "well the experts explained what they explained, on with the video" look, but then Grant asks another question, dogging in to the subject. I think this also kind of neatly demonstrates the difference between your channels, where Stand-up maths is more about exploring math and covering interesting new topics in an entertaining way every time, 3b1b is more about digging deeper and trying to get the viewer to understand one topic at a time.

Something important to note is the specific players that take long distance threes: most average three point shooters aren’t going to be taking 30 footers. The majority of those will come from guys like Steph curry, Damian lillard, all time level shooters. So there’s a good chance that inflates the points per shot the further out you go.

Love this - thanks for showing that data can be used for fun!

I love that you had Team Pi jerseys made.

I'm getting that delusional feeling that something (this video) was made just for me. Bec Hill and Grant Sanderson in a sports related Matt Parker video - really doesn't get more perfect than that.

Particularly enjoyed the discussion on stats vs coachable metrics, and some accommodation for "trusting your gut". Would also recommend sports+math people check out "Baron Davis From 89 Feet" which is a great story on an extreme outlier data point.

Thanks Matt, helps a lot!

Well this is a great crossover between my love for the NBA & Maths. Great stuff

You could explain the leveling off of the probability due to the interaction with the defense of the opposing team. Clearly it is also easier and more important to defend closer to the hoop. So this actually may be a game theoretical equilibrium where the advantage gained by going in deeper is negated by having to deal with a stronger defense.

Watching this video is a valid reason to not be revising for my AS Maths paper, right?

Super interesting. Loved this video.

This is awesome bc Ive loved Stand up matchs for years, and I've loved working with NBA data for years. Wish I could get a hold of these datasets

Grant makes very good points

As a maths and basketball fan I am so excited for this!!!!

I heard about this exact stat (how much each distance is "worth") and it's great to see it visualized. Also it makes more sense seeing it on the graph rather than just hearing numbers.

Where is the heat map of the whole court?

When I saw Matt's Team Math jersey, I really wanted to see the rest of the team labeled with other mathematical constants.

7:30, the NBA has these graphs (smoothed and shows frequency for each player on their shot chart page

Hot damn 3b1b is absolutetly ripped

Is "distances in INCHES" correct? 20 inches away and only 20% accurate?

I always wondered how the Globetrotter Planet would come to be, and why they'd be so good at maths..

Another part of the reason the trend toward more threes is so recent, is that the typical player in 2023 grew up in a world where it was well known that high volumes of 3 point shots were they way to go, and so they've spent a lot more time learning and training that skill. There were fewer skilled long-distance shooters in 1997 because when those players were growing up the 3 pointer was a new addition to the game and the skill wasn't really needed. That's especially true at the amateur level, since high school and college ball didn't adopt the 3 point line nationally til 7 years after the NBA.

Fascinating

Grant’s math jersey was a Parker Gift

Math and basketball and all that but anyone else notice Grants massive biceps? :P

How points-per-shot against distance would change if you had a new ratio of 5/4 instead of the current 3/2? In football-soccer they changed the victory to draw ratio from 2/1 to 3/1 and it worked!

"A single metric basketball player" This is now my new favorite unit.

18:37 Here's an interesting thought, would you be able to re-do that graph with the darkness of each point being a confidence level of the athletes who made the shot? Confidence being what portion of their shots are made from that distance.

Letting you know about errors in the comments, as is tradition: Labeled your own/everybody shot graphs as "Distance in inches", should be ft obviously. We'll cough it up to being in silly freedom units.

When I was at the APS March meeting this year I saw a great talk where someone used density functional theory to model basketball, basically treating the players as interacting particles. Lots of cool math you can do with sports!

Very interesting, thank you

You should have metric units on screen as well

Matt Parker x Jon Bois

I just realized that Matt's jersey shirts include a PARKER SQUARE(d) Well played, Math Parker, well played.

This is the kind of content I live for.

Soooo Grant is WAY bigger than his videos make him look.

5:13 wait…. Distance in inches? Hoping this was feet.

Great video!! I think the variance of the accuracy has a lot of importance too. Only looking at the expected value may distorce the strategies. Congratulations for the excellent work.

I love how nicely the 2D graph (points AVG vs distance) suggests that the shooting pattern (for optimal play) would end up looking exactly how the 2020 (3d visualisation) shot data looked.

Would have loved to have measurements in meters as well! Inches and feet are a little tough to parse

This is like that one Jon Bois video except on steroids. I'm in love.

i love the hint of the parker(square)

I’m a bit of a sports analytics junkie, so I had already seen all of this data a million times presented in a million ways, and I have to say that the data visualization in this video was really top-notch. In particular, the one showing yearly shot distribution plotted as 3d bars on a court overlay was probably the best presentation of that data I’ve ever seen.

Great video thanks!!

A huge thing to keep in mind is the strength of the defensive shot contest. I'm assuming the flat section of the shot-percentage/shot-distance curve in the mid range isnt actually about players being as good of shooters from 10 feet as they are from 20, but rather that opposing teams are much more happy to let a player walk into a deep mid range shot than a shot from floater range closer in, and so the defensive attention is a lot higher on the latter than the former, and then once the player gets out to 3 point range the defensive attention picks up. Defensive schemes like Drop pick'n'roll coverage explicitly aim to do this, first: making sure to fight hard through the screen to push ball handlers to within the three point line, and then once the ball handler is there, having the screener's man drop well into the paint to allow them to contest at the rim or those short mid range shots, with the goal being to concede semi-open long mid range shots.

Two of my favorite things - math and basketball.

So I’m a university student. I have finished calculus I and II and had a very very fun differential equations/linear algebra course. And while I love math, taking it out of theory myself is so tricky! seeing that 3-D plot around 4:30 was the tipping point though. I thought, "wouldn’t it be a neat metric to show the derivative of a linear fit of all the points at each position on a horizontal axis of time, to see how different places changed over time?" and felt like a math super genius lol. Thanks for the awesome vid!! I love math inspiration!!

The reveal of 3B1B playing in the background was so cool :p

I would love to see an average distance of shot as a function of year/season, which will more clearly answer the question than a heat map. I boldly claim.

Damn! Who knew Grant was so ripped? Also, I love that you can do anything you want with math, given enough creativity. I mean, since the Universe seems to run on math (or at least, that's the best way we have to describe the Universe) it makes sense. Yet another cool video! Thanks.

I did not expect to see this in my feed.

Thanks for showing us data can be use for fun.