Over the past couple weeks in this space I have explored the shot tendencies of the traditional positions. Â 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. Â PG Rajon Rondo shoots like a Center, PF Ersan Ilyasova shoots like a guard, and Dirk Nowitzki is in a class all in himself. Â No, these aren’t earth-shattering discoveries, not subjectively at least. Â 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. Â Does this sound right to you? Does the typical center have a small 3-point game? Â 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. Â 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. Â 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. Â 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. Â 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. Â *How many Justin Bieber songs have you listened to in the past 24 hours?*, your buddy asks. Â 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). Â 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. Â Sean collects the answers, smiles, and makes his big announcement: “The average guy in this room listened to Â 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). Â 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. Â 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. Â 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. Â 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. Â 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. Â 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. Â 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). Â You don’t see that range in the other positions. Â For centers, Brook Lopez represented the median value of 48 percent while the sample average was slightly lower at 46 percent. Â 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. Â 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. Â 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. Â The floater makes up most of the shots that a guard would take in this zone. Â They rarely have the chance to take a set shot or a post up in the further away in the paint. Â 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). Â 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. Â 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. Â Power forwards don’t like to shoot threes but they love taking long twos. Â Keeps them close for the rebound.

Pretty much all point guards feature a long two game. Â 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. Â (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. Â 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. Â Shooting guards are more tightly wound around the 30 percent median, but plenty of small forwards have little Â propensity to launch from downtown. Â In fact, half the small forwards shoot somewhere between 14 and 41 percent of their shots from beyond the arc. Â That’s their “interquartile range” from top of the box to the bottom, in case you were wondering. Â Mixed bag throughout.

In general, the wider the box, the more varied the shooters. Â On the flipside, a compact box indicates that there’s not much variance in the bulk of the distribution. Â 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. Â This probably indicates that there are wide variety of center subtypes at the rim. Â 3-point shooters, too, looking at the boxes for SF, SG, and PG.

Hopefully this reveals a little bit more about positional shooting tendencies. Â 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. Â I’ll look into it.