Monthly Archives: September 2010

Lopez Brothers Are All Relative

I’d like to be honest about something: the idea of identical twins freaks me out.

I know they’re a lot more common than I realize and I understand the science behind the process of making identical twins. However, there is just something about the actual visual and conceptual existence of identical twins that really scares me and leaves me feeling unsettled.

My deep-seeded consternation with identical twins may be pretty easy for me to go back and track. The Shining terrified me and it wasn’t because of anything but those creepy twin girls kicking it in the hallway with matching clothes. I didn’t mind the blood flooding the hotel walkways, the creepiness of Shelly Duval or Jack being a dull boy. It was the twins road-blocking the Redrum kid when he was just trying to big wheel his way through the Overlook Hotel.

Fast-forward many years and identical twins still give me the willies. And even as entertaining as Brook and Robin Lopez are on their own and especially in each other’s presence, I still can’t shake the uneasy feeling I get from two people whom look and act alike. The different lengths of hair don’t settle me either. Sure, you can tell them apart and their games are completely different with one being an offensive force and the other a defensive specialist. But the idea that they possibly have some level of ESP between each other and will always sort of be the same really bothers me.

However, thanks to Alby Einstein and the science community I may just be able to coexist in a world with the Lopez twins and their mutually exclusive identity. According to this Eryn Brown report from the LA Times, the theory of relativity is being proved true with “lasers” and the results are showing relativity can be scaled down to even smaller degrees:

Among the oft-repeated predictions of Albert Einstein’s famous theory of relativity is that if a twin travels through the cosmos on a high-speed rocket, when he returns to Earth he will be noticeably younger than the twin who stayed home.

Now physicists have demonstrated that the same is true even if the traveling twin is merely driving in a car about 20 mph. But in that case, when the twin gets home from the grocery store, he is only a tiny fraction of a nanosecond younger, according to a report in Friday’s edition of the journal Science.

Now, I’m no science major. In fact, I struggled in science. But to me this sounds like the more a person travels and the faster a person travels while the other stays more still, the younger this “moving” person will become in relation to the stationary person. Perhaps this isn’t the most sophisticated way of explaining part of the theory of relativity – and yes, I would expect Albert Einstein is furious with me right now – but that’s essentially what this study was saying.

If this theory is true, and I believe science is telling me that it is, then the theory of relativity will help my uneasiness with the Lopez twins. Even though Brook and Robin have different hair, different uniforms and probably different versions of Thor that they enjoy, they’re still so identical that it creeps me out. It gives me some solace to know that Brook is an offensive force while Robin is the answer to many of the Suns’ prayers for a defensive presence in the middle. I’m fascinated at the idea that the two of them excelled at very different parts of the game that probably heightened their skills even more.

Brook probably became such a good offensive player because he had to score against Robin who was so good at defending. Or was it the other way around? Did Robin become such a good defender because he had to figure out how to stop Brook from dominating him in the driveway? There is something very chicken or the egg about this.

Regardless of how or why their particular set of skills got honed, the differences between Brook and Robin are going to increase over time with the current organizational philosophies of their respective teams.

The Phoenix Suns have been near the top of NBA pace over the last several years. Even with coaching changes and personnel being switched out like faulty spark plugs, they remain amongst the fastest teams in the league. This past season, they were fourth in the NBA in pace at 95.3 possessions per game. The New Jersey Nets on the other hand were quite slow with 91.4 possessions per game (good for 24th in the league). And with Avery Johnson taking over as they begin their transition from meadowlands to Brooklyn, they probably won’t get any faster out there. During his three full seasons coaching the Dallas Mavericks, Avery kept his teams at the 27th, 28th and 24th fastest paces in the league.

This means that over time Robin Lopez has found his way into an identical twin NBA fountain of youth. Robin will always be moving at a much faster pace assuming his situation and Brook’s situations remain fairly constant. Stick with the run’n’gun style of the modified SSOL and Robin should enjoy the benefits of a stylistic anti-aging cream. On the flip side of that, Brook and his franchise’s refusal to get out and stretch their legs a bit will probably age at a quicker pace.

However, there might be something to level the playing field for Brook – his franchise’s location. While the speed of the game for both of these teams seems to favor the bang for Robin’s buck throughout his career (compared to Brook’s), the location of these players may even things up.

The reverse is often said to be true for a twin who spends time high on a mountaintop; general relativity predicts that time passes more quickly at greater altitudes because objects don’t feel Earth’s gravity quite as strongly. But the physicists found that a twin who lives just about a foot above sea level will age ever-so-slightly faster than a twin living at sea level.

The city of Phoenix, Arizona sits roughly at an elevation of 1,117 feet. The city of Newark, New Jersey resides around 30 feet above sea level. So while Robin can run around and stay younger all he wants, Brook’s ability to ball close to the level of the ocean CAN have an affect on how he ages in relation to his twin. While this sounds like Brook can turn to his brother’s Benjamin Button style of play with a “take that!” in reality the elevation factor may not be enough to truly matter. According to the paper, “the second hand of a clock positioned about two-thirds of a mile above an identical clock near Earth’s surface will speed up only enough to tick out three extra seconds over the course of a million years.”

As good as Brook Lopez is he probably won’t play for a million years. Considering Kevin Willis was a big man modern marvel by playing into nearly his mid-40s, that’s asking a lot to think Brook could 7-figures in terms of the length of his career.

In the end, the Suns’ ability to let Robin play at a high pace definitely makes the duration of his career seem to be worth it more than the location of Brook’s home floor does for his longevity.

And while you’re probably wondering why you just read through this entire article and learned virtually nothing, we did learn a few key things:

1. I know next to nothing about science.
2. Sometimes, it’s good to stretch your legs a bit and delve into a subject you don’t understand.
3. Identical twins really freak me out.
4. NBA media day is today and that means training camp begins tomorrow.

Welcome back, NBA season!

NBA HD: MORE Positional Analysis

Over the past couple weeks in this space I have explored the shot tendencies of the traditional positions.  I’ve set out to find the players who don’t shoot like their designated positions by comparing their shot location distribution to the average shot location distribution of their conventional position.

I’ve found some pretty interesting results. Point guards, shooting guards, and small forwards have nearly indistinguishable average shot distributions.  PG Rajon Rondo shoots like a Center, PF Ersan Ilyasova shoots like a guard, and Dirk Nowitzki is in a class all in himself.  No, these aren’t earth-shattering discoveries, not subjectively at least.  Statistically maybe.

I think these were worthwhile exercises but I wasn’t fully satisfied with the method I used to look at the shot distributions.

Why? Look at the average shot distribution for a center in the table below:

What this tells us is that 6 percent of the average center’s shot distribution comes from beyond the arc.  Does this sound right to you? Does the typical center have a small 3-point game?  With all the stretching that goes on with today’s big men, it may seem as though 3-point shooting centers are on the rise.  Channing Frye, Brad Miller, Rasheed Wallace, Andrea Bargnani, and Mehmet Okur are conventionally considered centers but they make up just a small subset of the center position.  I’d venture to guess that half of the NBA’s centers shot as many NBA 3-pointers last year as you or I did: zero.  So why does this table tell us that the average center shot a healthy dose of threes last year?

What we’re dealing with is a skewed sample distribution.  Not in shot location sense of “distribution” but the skewed shape of the underlying data.

Say you’re hanging out with your five of your buddies when one of them decides to take a poll.  How many Justin Bieber songs have you listened to in the past 24 hours?, your buddy asks.  The five of you write down their answers on a sheet of paper and hand them to the Bieber-curious poll administrator (we’ll call him Sean).  You, Pat, Steve, and Matt confidently write down that they’ve listened to exactly zero Justin Bieber songs. But Sean? Sean loves Justin Bieber. He confesses that he has listened to 10 Justin Bieber songs.  Sean collects the answers, smiles, and makes his big announcement: “The average guy in this room listened to  not one, but TWO Justin Bieber songs in the past 24 hours. I knew it! I’m not the only one!”

You see what Sean did there? He added up all the total songs listened by the group (10) and divided by the number of members in the group (5).  But Sean’s obsession skews the distribution and the sample average (mean) misrepresents the general song tastes of the group.

In cases of skewed distributions, it’s better to look at the sample median rather than the sample average (mean) because medians are less sensitive to the extremes.  For those who skipped the mean, median, and mode portion of fourth grade, the median tells us the middle observation in a sample.

Back to basketball.  The typical center didn’t shoot 6 percent of his shots from downtown, just as the typical buddy didn’t listen to 2 Justin Bieber songs.  Twenty regular centers in the 41 person sample didn’t even take a 3-pointer last year, making the median 3-point share among centers last season 0.5% from Clippers center Chris Kaman. That’s an important tweak.

So I wanted to run this tweak for all the positions and illustrate their distributions. I decided to display the data in the form of a box plot, also known as a box and whisker plot.   These plots pack a ton of information in a nice tidy graph: the median value for the sample, the smallest value, the largest value, where the bulk of the observations fell, and, if they exist, outliers in the sample.

Let’s take a look at one here that illustrates the at rim shot distributions of each traditional position:

So here we have 5 box plots. The top of the box tells us where the 75th percentile observation lies and the bottom of the box tells us the 25th percentile observation.  The box therefore, represents where half the distribution lies. The line in the middle? That’s the median value. I would have displayed the average line to give you an idea how the two descriptive statistics differ but I didn’t want to confuse. You can see the average values in the table above.  The lines that stick out are the whiskers, detailing the maximum and the minimum of the sample.

So what do we learn? Centers have the widest range of at rim tastes, both in the box and the whiskers. You have Channing Frye with 10.9 percent share at the bottom and Joel Przybilla who never ventures away from the basket (94.0 percent of his shots were layups/dunks).  You don’t see that range in the other positions.  For centers, Brook Lopez represented the median value of 48 percent while the sample average was slightly lower at 46 percent.  We can confidently say that layups and dunks make up half the shots of a typical center.

The other positions are more tightly packed, indicating that centers are unique in their varied shot tastes. Or it’s the other way around: centers are the most heterogeneous position because their center label has the least to do with their playing style. Who’s the tallest guy on the court? He’s the center.

What’s also interesting is that positions descend in their taste for shots at the basket from center to shooting guard but point guards jump up a bit.  Why? This is just my take but point guards are usually the quickest on the court and run the offense, and therefore can get to high-percentage spots more often than their taller teammates.  It’s hard to get a layup when you don’t have the ball in your hands or the quickness to evade defenders.

You’ll notice an upper outlier for both small forwards (Gerald Wallace) and shooting guards (Ronnie Brewer). We should flag these guys as players who probably don’t fit their positional label since they certainly don’t shoot like it.

Let’s move on to “short” shots, the attempts that were less than ten feet from the basket but not layups or dunks.

Here we see that guards rarely get a shot off in this zone and shooting guards especially have a compact distribution.  The floater makes up most of the shots that a guard would take in this zone.  They rarely have the chance to take a set shot or a post up in the further away in the paint.  Also, the positional “shape” mirrors the last zone where the shot taste descends to shooting guards from centers, with point guards exhibiting a slight up-tick.

Let’s take a look at the mid-range area which is 10-15 feet from the basket.

Not much doing here. Only five players shot over 20 percent of their field goal attempts from this area last season (Elton Brand, Shaun Livingston, LaMarcus Aldridge, Dirk Nowitzki, and Rip Hamilton).  Moving on.

Long twos:

Note that centers have the widest distribution as well as the lowest median, while shooting guards have the tightest distribution and the highest median value.  Power forwards and shooting guards have similar medians but a larger number of power forwards make long twos a big part of their shot palette.  Power forwards don’t like to shoot threes but they love taking long twos.  Keeps them close for the rebound.

Pretty much all point guards feature a long two game.  The most long-two resistant point guard last year was Chris Duhon and even he took more than the typical center did. Gotta have that pull-up jumper to keep the defender honest off the dribble.  (FYI, one out of every five of Rondo’s shots are from the long two zone despite shooting just 33 percent from there).

Let’s glance beyond the arc:

This is what I expected. Most centers have no 3-point game to speak of but the statistical mean suggested that the typical center has shoots a three once every 19 shots.  Power forwards, too.

Most wings have at least 30 percent of their shots coming from beyond the arc. Not exactly mind-blowing but shooting guards exhibit a much more compact box than small forwards. What does this mean? Small forwards are a mixed bag when it comes to 3-point shooting. You have Shawn Marion who barely shoots threes and you have James Posey who loooves shooting from downtown.  Shooting guards are more tightly wound around the 30 percent median, but plenty of small forwards have little  propensity to launch from downtown.  In fact, half the small forwards shoot somewhere between 14 and 41 percent of their shots from beyond the arc.  That’s their “interquartile range” from top of the box to the bottom, in case you were wondering.  Mixed bag throughout.

In general, the wider the box, the more varied the shooters.  On the flipside, a compact box indicates that there’s not much variance in the bulk of the distribution.  The biggest interquartile range of the bunch? Centers at the rim. You can’t definitively say, “A typical center should shoot X amount at the rim” when the distribution is so dispersed.  This probably indicates that there are wide variety of center subtypes at the rim.  3-point shooters, too, looking at the boxes for SF, SG, and PG.

Hopefully this reveals a little bit more about positional shooting tendencies.  It’s not the averages we should be so concerned with, but the distribution.

I’m currently working on some k-means cluster and PCA statistical analyses that I think will blow the lid off the positional revolution. As is, they’re not quite ready to publish yet. Consider this is an appetizer.

Lastly, I’m not sure why some of the outliers are wonky. I’m using a program to spit out these charts and some of the outliers pretty much sat on the whisker ends.  I’ll look into it.

NBA HD: Positional Identity Crises Part II

Earlier this year, I published a post at Hoopdata that analyzed player’s who shoot like a position unlike their own.  (Too bad a virus gobbled up the article archive or else I’d link to it.) In that piece, Rajon Rondo, Channing Frye, and Kobe Bryant were featured as players who were contrarian shooters. Today, I’d like to update that post with a more rigorous statistical technique, z-scores, that I used in last week’s post.

So here’s what I’m asking:

Which players shoot like a particular position who are not actually members of that position?

As a refresher, here are the summary stats from my positional analysis.

I’ve noticed that the shot location makeup of point guards, shooting guards, and small forwards are very similar.  Take a look at their percentages under the share column.  Almost identical, right? What distinguishes the three positions is not their shot densities but their shot types (spot-up, dribble-drive, off-screen) and shot source (assisted or not).   You’ll noticed that on shots at the rim (layups and dunks) small forwards get assisted twice as often as point guards (55.5 percent vs. 28.4 percent).  The reason is obvious: point guards are usually the ones feeding not the ones being fed.  So, even though the shot location is in the same area, it’s a different kind of shot.

Why do I bring this up? You’ll see a lot of overlap in the next few tables.  Since the makeups are so similar, you’ll notice the same players will keep popping up for multiple positions.  O.J. Mayo was the most shooting guard shooting shooting guard from last week’s post and likewise, he shoots a lot like a point guard and small forward.

Nonetheless, you should learn a thing or two from these tables.

Some ideas for next time:

- Using median shot location shares instead of average shot location shares to correct some skew issues. For example, does the typical center shoot 6% (the statistical mean) or does the typical center shoot none (the statistical median).

- Since PG=SG=SF, separate into two groups: guards (PG, SG, SF) and bigs (PF, C). Lose some of the detail but more instructive.

- Player comps rather than positional comps.  Since Dirk Nowitzi doesn’t really ascribe to a traditional position, how about trying to find players who shoot like Dirk?  Suggestions for player comps are welcome.

- K-means clustering.  DSMok1 with another fantastic suggestion in last week’s comment section, asks if we could do a k-means cluster analysis.  I think we could, although it’s not my statistical proficiency.

- Heat maps using r, courtesy of chart genies Albert Lyu and Jeremy Greenhouse.

Hit me up on Twitter at @tomhaberstroh if you have any other ideas or just comment below as usual.

NBA HD: Visualizing Shot Selection by Position

The positional revolution has gained a full head of steam over the past month.  Although talk of tearing down the walls of traditional positions has been going on for years, Drew Cannon’s brilliant article at Basketball Prospectus blew the discussion wide open and sparked a slew of articles from the game’s brightest writers and analysts.

Definition is the root of the issue.  What is a point guard? Besides height, what differentiates a power forward from a center?  Why do we call a player who can’t shoot a lick a shooting guard?

Here’s an attempt at quantifying those definitions from the shot selection standpoint. Using’s player shot location data, I’ve calculated the average shot location shares of each position (the positional designations on Hoopdata come from

We want to outgrow the conventions of traditional positions but let’s see if we can observe divisions in the first place.  Hoopdata breaks down shot types into 5 buckets: at the rim (layups and dunks), <10 feet, 10-15 feet, 16-23 feet, and 3-point shots.  Here’s how the five positions look, in terms of percentage of shots in each location.  So what does a point guard’s shot makeup look like compared to a shooting guard? Where do we see the biggest disparities between positions?

Here we see that the typical point guard attacks the basket more than the typical shooting guard and then the basket attack trends upward with the following positions.  Most point guards work out of the pick-and-roll which lends itself to penetration to the rack or dishes to the rolling big.  They’re getting almost all of their at rim baskets on penetration as opposed to bigs who can get layups/dunks from offensive rebounds.

Looking further, we see that the mid-range jumper is the least populated area for shots but there isn’t much distinction between positions in the mid-range.  What’s also interesting is that point guards, shooting guards, small forwards, and power forwards all shoot the long two in similar doses, with centers only taking about 18 percent of their overall game from here.

From the 3-point line, it makes sense that shooting guards launch the most from deep and the centers the least.  Nothing too ground-breaking there.

Perhaps what’s most interesting is how similar point guards and small forwards are in their shot palette.  The blue and green bars are nearly identical with each other from zone to zone.  Below is the graph in table form along with the assisted percentages and field goal percentages from each shot location, courtesy of Hoopdata.

There’s plenty of good stuff in the table above but for now, let’s dig deeper and see which players get classified in a particular position but shoot nothing like their traditional brethren.  To get there, I calculated each player’s z-score (which, in simple terms, calculates the magnitude of deviation from the norm) compared to the positional mean from each shot location.  Then, I took the absolute value of those z-scores and summed each location together for the Zsum to get the final aggregated score.  Note: I only looked at players who averaged 20 MPG and played 20 games last season.

In the first table below, we find that Miami Heat point guard Carlos Arroyo deviates the most from the shot selection of a traditional point guard.  In particular, 65.3 percent of his shots come from long twos and he barely attacks the basket or launches from downtown.  His z-scores total to 8.19 which is the highest sum of the point guard bunch.  Perhaps is good that he doesn’t attack the basket, as he only converts on 47.8 percent of his tries which is far below new Charlotte Bobcat Shaun Livingston’s 71.4 percent success rate.

The first table displays the “Least Alike” players in the group and the next table shows the “Most Alike” which tells us who are the most protypical point guards in their shot selection.  Orlando point guard Jameer Nelson tops that list.  I’ll save the commentary for a later date but I found this to be a pretty interesting exercise.  Which players are positional contrarians? Find out below. (My apologies for the blurriness).





Blog-on-Blog Love

If you’ll excuse me, I have some blog doting to do.

M. Haubs and Jay Aych have been absolutely killing it with their previews and coverage of the FIBA World Championships at The Painted Area. This is nothing new. TPA has been an established go-to source for both NBA and international basketball content for some time, and the only thing that gives me more pleasure than taking in one of Haubs’ or Aych’s fantastic pieces is seeing them get their due.

And a plug by Fran Fraschilla on today’s Team USA broadcast? Not too shabby.

Fraschilla: “You know, Mark, I love reading the blogs. We have a guy, Jay Aych, The Painted Area, great blog this week. (I hope I said his name right.) But he made a great point about teams in Europe and international play, they run their offense like the Utah Jazz run it – very crisp, disciplined, lot of touches. And that’s why the Jazz give teams in the league – you know, Jerry Sloan’s team gives teams so much trouble. Team USA is not used to guarding all 24 seconds of the shot clock, in my opinion.”

For reference, Fraschilla was referencing this post by Aych following USA’s win over Brazil:

This is not a surprise as NBA players don’t see that type of off-ball movement in the U.S.–lots more moving parts to deal with in FIBA ball. Brazil ran a lot of continuity sets, like the ones Magnano’s Argentina teams used to perplex Team USA with. It’s not just pick/roll that befuddles Team USA, it’s the off-ball action and screens coming from all angles. You will see a lot of variations on Princeton sets or flex sets in this tourney. Constant offensive motion is a staple of int’l basketball. Offenses with reads, counterplays, and counterplays to the counterplays.

If we were preparing Team USA for what to expect in this tourney, we’d tell them that it’s like playing the Jazz many times. And if you ask NBA players about defending the Jazz offense, we’re sure most would say it’s not fun. (More on this topic as it relates to Team USA, from ’07.)