The Algebra of Scoring and The Calculus of Keeping Score

One of the first things a pickup baller does when coming to a new court (after they’ve found out who’s got next and if they have five) is to find out what scoring system they use. Watch a few possessions and you know if the cats can play, watch a little while longer and you notice who can shoot, and who (if anyone) passes. You can tell if they prefer turn-taking, clear-out, iso-style pickup–the cognitive equivalent of tic-tac-toe and checkers or if their approach is more strategic–say like chess or go. Do they collectively treat pickup like a sprint, like a marathon, like a brawl, or more like a picnic? These are all things one can get a sense of after just a few minutes of observation, yet, scoring is a parameter of play that’s better asked about up front. Not only does knowing if games go to 11, 12, 15, 21, or 30 help you estimate if you’ve got enough time to go take a Bobby Hurley before your game starts, it ensures that you’ll avoid being seen as a total newbie on the court, and it enables the application of the correct end-game strategies.

While pickup is an activity that gives its participants a sense of ‘having fun,’ most regular pickup ballers equate fun with winning–thus the importance of keeping score. Knowing the scoring intervals: ones and twos (1sN2s) or twos and threes (2sN3s) combines with the score to impact the micro and macro elements of strategic late-game play that can mean the difference between winning or watching the next game from the sidelines.

Unlike Josh Waitzkin, who offers his opponent a draw, pickup ballers play to win.

Having played regular pickup ball in Austin, Boston, Champaign, Houston, Mankato, Mazatlan, and Quito I’ve experienced a number of different pickup cultures, each with its own system of scoring. With the exception of the ARC courts on the University of Illinois Urbana Champaign campus (where the winning team decides between two different scoring options), most places have a set scoring system. Over time, I’ve built up some theories about the pros, cons, and quirks of each system but I hadn’t explored these in any depth until this past week when I started adding formulas to an Excel spreadsheet and tinkering with some sort of comparative analysis (I thought about doing it in Python but Excel won out due to *prettier* screenshot potential). So in the name of better, more strategic basketball, I give you a math-based analysis of scoring systems with the end-goal of increasing your pickup knowledge and the frequency with which you stand victorious.

How pickup ballers feel after stringing together 6 or 7 wins in a row.

Personally, I find the social/qualitative aspects of pickup basketball are more interesting than trying to quantify those aspects in hopes of getting a paper accepted at the MIT Sloan Sports Analytics Conference (this would be difficult because, as I explain later, the statistical data on pickup ball is practically non-existent). The social sciences are rife with well-meaning knuckleheads trying to turn things that are more complicated than categories or quantities–things like knowledge, feelings, and learning into numbers. This is a problem in many fields, yet sometimes containing or collapsing complexity into numeric forms in hopes of making sense of things on a grand scale appears to be the only way forward.

In the case of systems of scoring in pickup basketball it makes complete sense to apply mathematical thinking to factors that are by nature, already numerically based. So applying quantitative analysis to systems of scoring pickup basketball passes the knucklehead test.

In the sections below I explain how I came up with the frequencies and percentages I’ve used in the calculations so others can replicate, extend, and/or critique them. Depending on where you work and what you do being seen doing the math may position you as part of the Matherati who (a) should be avoided or (b) are going places and deserve admiration.

Don't panic but this post is computationally intensive
Don’t panic but this post is computationally intensive

We begin our journey with the one provable statistic we have in pickup basketball, namely the score. Pickup ball doesn’t have a stats department, it’s a culture with no administrative overhead and very little organizational reflection. People come and shoot around until it’s their turn to play, they might watch a few plays of the game before theirs to get a sense of the competition, they might ohh and ahh over an in-game dunk or no-look pass but that’s pretty much it. They’re not going to keep track of how many times the winning and losing teams shoot from inside or chuck it from beyond the arc. They aren’t gonna tally up the number of possessions or log  per-player turnovers.

What’s more, even if you found someone to keep stats for a game or two, pickup basketball is so varied that it would take data from hundreds if not thousands of games at dozens of locations to get any rigorous sense of anything. Given our surveillance-culture, this might become possible at some point when security cameras pointed at basketball courts are combined with automated stat-keeping applications by some pickup-baller who also works as a NSA analyst. Until then, pickup ballers and pickup culture chroniclers are left with intuition and accrued sensibilities about a range of basketball-related metrics.

Personally, when I play ball I’m in the moment. I’m not thinking about anything except the current play. That’s part of the beauty of pickup ball, it’s a way of resetting stress levels and clearing one’s mind. It helps block out the off-court world for a few minutes or hours. Cognitive psychology tells us that humans can keep 1-10 pieces of information or 1-4 chunks of knowledge in short-term memory at any given time ( As pickup ballers we’re constantly thinking about so many dynamic in-game factors that even the score of the game we’re playing in can get pushed out of short-term memory. Most games have at least one in-game stoppage wherein teams try and remember who scored, which team is leading, and why the other team isn’t actually up by two like they say they are. This pretty much rules out keeping track of in-game shooting percentages, turnover rates, and shot distribution.

Forgetting the score is not limited to pickup basketball.

While each court may vary in terms of how many points are needed to win (11, 12, 15, 21, 30) and what scoring intervals are used to get there (1sN2s or 2sN3s), all courts do have an endpoint (either fixed or win-by-two). So let’s begin with the cold hard numeric facts of pickup basketball and see what, if anything, they tell us.

Part I: Scoring Intervals and Thresholds

What is the range, minimum, and maximum number of FGs needed to win at the different scoring levels?

In 2003, the NBA changed the first round of the playoffs from a best of five series to a best of seven ( While increased revenue was surely a factor in making the switch (random people on the Yahoo message boards insist that the unpublished per-game playoff revenue to be between 1.25 and 2.45 million), the NBA’s public rationale for going to a seven game series for the first round was because they felt the longer the series, the more likely the superior team would advance to the next round. The league stated that this change would ensure better basketball throughout the playoffs and not overly reward atypical hot shooting.

Who knows if the 1994 first round series between the 8-seed Nuggets and the 1-seed Supersonics would have ended with Dikembe celebrating if the series had been a best of seven instead of a best of five. [Photo:]
More play being an advantage to the better team is also true within a single game. There’s a reason each successive level of competition plays longer games. Most high school games are played using 8 minute quarters resulting in 32 total minutes of play, college games use two 20 minute halves for 40 mintues of game play and NBA games employ 12 minute quarters–yielding 48 minute games. The progressively shorter shot clock is designed to create more possessions which increases watch-ability while the added time is meant to improve the chances that a team’s performance will have less to do with luck and regress or stay somewhere close to their mean or average ability level (Lowe, 2013).

Pickup ball not having a time limit, requires a reliance on reaching a certain score to determine a winner. Any 10 pickup ballers about to begin would happily play a competitive game to 45 by 1sN2s. This of course would make it impossible for anyone else to get on the court. Some games to 15 by 1sN2s can take up to 50 minutes to play as it is, imagine what it’d be like if each game took 2 to 3 hours to play? People would end up calling two down before going to the library to study for 4 hours before returning to play their game against a very tired opponent.

I don’t expect any of the findings in this post to influence on-court scoring systems but I do have some biases about what systems do the best job at preserving the general strategic elements of balanced competitive basketball. In the body of this post I’ll look at the different systems already in place to get a sense of how they compare to each other. Once all the comparisons have been made I’ll critique elements of different scoring systems in the conclusion. The comparison will include the following scoring systems:

  • Games to 11 (only 1s) -Outdoor courts: e.g. Parque Carolina Quito Ecuador
  • Games to 11 (1sN2s) -Outdoor: Parque Martiniano Carvajal Mazatlan Mexico
  • Games to 12 (1sN2s) -Outdoor: Root Memorial outside Houston’s Toyota Center
  • Games to 15 (1sN2s) -Indoor courts: Option 1 at the ARC Gym at UIUC
  • Games to 21 (2sN3s) -Indoor: Option 2 at the ARC Gym at UIUC + Noon Games
  • Games to 30 (2sN3s) -Indoor: Gregory Gym at UT Austin

Except for the single-interval purists who play to 11 and give 1 point for all made FGs, there’s a range of total made field goals that add up too a winning score. The table below shows this range organized by scoring system.

The range of made field goals required to win based on different pickup scoring systems
The range of made field goals required to win based on different pickup scoring systems

As one might expect, playing to 11 or 12 by 1sN2s requires the fewest FGs to win while playing to 30 by 2sN3s and 15 by 1sN2s requires the most.

The range suggests a greater variance in the number of FGs needed to win based on possible FGAs at different combinations of scoring intervals. Making FGs from beyond the arc as opposed to inside the arc matters more in games played with 1sN2s than in games played using 2sN3s as scoring intervals–this is especially so when the game is played to a higher number. Conversely, the fewer the points needed to win, the smaller the FG range (especially when the scoring intervals are closer together (2sN3s)).

As mentioned before, a rather curious quirk of the courts at the ARC gym at UIUC is that the winning team gets to choose if they want to play to 21 by 2sN3s or to 15 by 1sN2s. Previously, I would have thought most teams choosing to play to 21 by 2sN3s did so primarily because they felt they didn’t shoot as well from beyond the arc as the other team. After looking at the FG range, it seems like there are several different reasons a team might choose to play to 21 by 2sN3s.

  1. Everyone agrees to play shorter games because it’s the beginning of the semester and the courts are clogged with freshmen who have yet to fail their first exam.
  2. The alpha baller on the winning team has a girlfriend who is watching so he wants to rack up as many wins as possible.
  3. The winning team doesn’t think they’ll shoot as well from beyond the arc as the squad who has next.
  4. The winning team doesn’t think they’re as good as the challengers but may be able to win a shorter game before the other team settles into a rhythm or recovers from cold shooting due to a lack of warm-up time.
  5. The winners are old-school (they also keep suggesting teams play shirts vs skins)
  6. The winners usually play in a Noon Game and need to get back to work in exactly 50 minutes.

One more thing about scoring intervals and FGs made. Unless you shoot FGs from beyond the arc as well as you shoot from within the arc, whether the court plays by 1sN2s or 2sN3s there are times when shooting from beyond the arc doesn’t offer any advantage. For instance if a team has 17 points in a game to 21 by 2sN3s they’ll need two FGs no matter if they’re shooting layups or from way out by the drinking fountains.

While this isn’t exactly breaking news, it denotes a pattern. Making an odd number of FGs from beyond the arc when playing by 2sN3s to an odd winning score like 21 or making an even number of FGs from beyond the arc is optimal when playing to say 30 by 2sN3s is more efficient. Likewise, in games scored by 1sN2s, an optimal number of FGs from inside the arc coincides with the winning or target score being odd or even. In other words, it’s a simple pattern related to overage. Making 6 FGs from beyond the arc is more than you need in a game to 11 by 1sN2s as you waste the final points. Likewise making an odd number of FGs from beyond the arc in a game to 30 by 2sN3s or an even number of FGs from beyond the arc in a game to 21 by 2sN3s results in a wasted FG from a longer distance which often means a lower shooting percentage. .

Patterns are more easily spotted as the game progresses.

How can we use the above scoring system data to theorize about other statistical phenomena?

From this point on we shall be leaving the firm foundation of fact and journeying together through the murky marshes of memory into thickets of wildest guesswork. -Dumbledore in JK Rowling’s HBP

As the quote suggests, once we move beyond FGs needed to win, we find ourselves in a bit of a guessing game due to the volatile nature of pickup ball. I’m not talking about conflict or fighting but rather, the unpredictability of the play. While there are some ballers who play often enough and/or are just plain good enough to have consistently good games, the most consistent thing about pickup ballers is their inconsistent play. For instance, I’ll have a few games where I’m leading the fellas like an inspired, knows-what-he’s-doing Jackie Moon (see clip below starting at 1:48).

Jackie Moon at his basketball best.

But then, I’ll have a handful of games where I mostly rebound and play off the ball despite being a capable lead guard (imagine a less dirty Reggie Evans). I’ll follow that up with a few games where I play like Austin Rivers in his rookie NBA season or Manu during most of the 2013 Finals.

In 2012-2013 Austin Rivers had one of the worst seasons in the history of the league [mashed-up image]
In 2012-2013 Austin Rivers had one of the worst seasons statistically in the history of the league [mashed-up stats and image]
So until the Edward Snowden of pickup basketball comes along and gives us the secret data, we’re left with educated guesses about: beyond the arc tendencies, shooting %, shot distribution, turnover rate, number of possessions, not to mention True Shooting percentage, PER, WARP, and BPRs. But we do have a wealth of NBA and college basketball statistics. It might be useful to estimate and extrapolate based on years of observation and comparative values within these well-chronicled basketball cultures to see what such inquiry might uncover.

For this and other stats I’ll start with whatever Google returns, in some cases it’ll be league-wide averages at the NBA and NCAA levels for the past season. When I don’t have access to league-wide averages I’ll cherry pick a team. Once or twice, data from High School players is included. When nothing else comes up, I’ll quote random dudes on the Yahoo! message boards. The idea is to start with actual numbers so that the estimates that follow are based in something more than pure intuition.

Brian Fantana sums up the precision of estimated statistics.

What is a ballpark pickup basketball field goal shooting percentage?

In 2012-2013 the Love-less T-Wolves shot 44% from the field and 31% from behind the arc ( while the Big 10’s Cornhuskers shot 40% and 30% respectively ( While these teams represent some of the lowest levels of achievement in their respective leagues, we can only expect pickup ballers to collectively shoot at a still-lower level of proficiency.

In a test of two thousand HS athletes’ jump shot execution, only 9% of them were found to have the proper angle and depth of entry over the rim. This lack of optimal depth and angle (correctable for the low low price of $3999) is said to reduce shooting percentage. We can only assume that few pickup ballers are among the Chris Mullin-like 9% who have optimal arc and depth. Anticipating that pickup ballers will collectively shoot a lower percentage than NBA and NCAA players, we’ll use the disparity between the pro and college averages as a point of departure in calculating inside the arc and beyond the arc FG%.

The geometry of the jumpshot [photo:]
Ray Allen using the NOAH during practice.

So if we deprecate Nebraska’s FG% by the difference between it and the T-Pup’s FG% we get 36% on FGs inside the arc and 29% outside the arc. However this crude Pro-College adjustment is glaringly problematic in a minimum of two ways. First, comparing FG% beyond arcs set at different distances is questionable, and second, unless you’re the Bobcats or Andrew Bynum, pro and college teams and players work hard to ensure that only the best shooters take the more distant shots. In pickup ball, unless a team has played with each other several times or has an uncanny feel for their roles, everyone, including the guy in the neon tank top who just finished lifting weights, feels entitled to chuck a few from the cheap seats.

For example, a few days ago I was playing on a team that had a guy who was money from inside the paint (4 for 5) and yet he kept drifting outside to unapologetically heave set-shots from deep with a conviction that would have made Kobe proud.  He went something like 0 for 8 from behind the arc and we lost (I’m not saying that he alone cost us the game, but his terrible shot selection really hurt us as he was also our best rebounder).

NBA Guard Propensity to Shoot or Pass also Plotted by Usage Rate [Photo: 2011]
I tried to encourage him to keep posting up but after he bricked his 6th three I asked the guy he came with if he could get him to stop taking 3s. ‘He won’t listen, he just wants to shoot,’ he told me. When I suggested to Mr. Chitwood that maybe he wasn’t a three-point shooter he told me to, ‘shut the ***k up‘ and promptly clanked another pull-up from deep–simultaneously proving his right to shoot, his irrational confidence, and his utter lack of shooting prowess. It’s important to have confidence but it’s also a good idea to work together based on a winning strategy.

Confidence is important, especially when paired with actual skill.

While pickup ballers usually have a bit more of a conscience than the guy I described above, there’s less discipline in terms of who shoots when, from where, and under what conditions. If teammates are missing shots from 2-6′ I’m generally supportive as they at least managed to take higher percentage shots. But if they brick a bunch from deep I tend feel the game potentially slipping away, especially if there are other people on the team who are actually making shots.

In 2012-2013 NBA centers shot just 10 percentage points lower on three pointers than guards, yet they took less than 1% of all the FGs taken from beyond the arc (see table below based on data from

Beyond the arc shooting data by position for the NBA 2012-2013 season
Beyond the arc shooting data by position for the NBA 2012-2013 season

Given the less-strategic, more-democratic distribution of shot taking; adjusting shooting percentages downward yeilds an indoor estimated inside the arc FG% of 34% and a beyond the arc FG% of 21% (I adjusted 10 percentage points downward by subtracting 2% points from inside the arc and 8% points from the beyond the arc).

What about outdoor shooting percentages?

Playing outdoors requires a whole range of coping strategies, from playing on oddly slippery surfaces, to extreme temperatures, to cockroach migrations, burned retinas, humidity, sea breezes, and gale-force winds.

In the history of pro and collegiate basketball there are very few outdoor games. The Suns have played in three preseason games in the past five years while college basketball has had two NCAA Men’s games played under the Carrier Classic monicker.

After beating #20 San Diego State University on a aircraft carrier Syracuse coach Jim Boeheim said this of coaching his team in the Carrier Classic, “This was a tough day, this was a hard game. You had to get to the basket. That was the only way you were going to score. You weren’t going to make any jump shots.”

NBA preseason games played outdoors have averaged 41% from inside the arc and 26% from deep ( Meanwhile, college athletes playing on the decks of ships have collectively shot 42% from inside the arc and 14.8% from beyond it (, Informed by these numbers let’s reduce the outdoor pickup FG% slightly from 34% to 31% for FGAs from inside the arc and more drastically from 21% to 13% from beyond the arc (again an overall 10 percentage point drop but just 1.8% below the collegiate percentage from deep).

To be sure, there are indoor and outdoor pickup games played with skill, strategy, and restraint by teams that employ set roles that include shooters who convert at levels well above these estimates, yet collectively indoor and outdoor pickup basketball is often a master class in rebounding.

What is the average number of field goal attempts needed to win from inside and beyond the arc?

Based on the shooting percentages established above and plugged into the table below, we see that shooting exclusively from inside the arc is 1 to 3.5 FGAs more efficient when scoring with 2s and 3s (to 21 and 30 respectively). Conversely, when scoring with 1s and 2s (to 15), shooting solely from beyond the arc is 6 FGAs more efficient than shooting only from inside the arc. Meanwhile, the outdoor penalty for shooting from deep is between 8-10 FGAs per game.

Average number of field goal attempts needed to win shooting exclusively from inside or beyond the arc based on a set team average field goal percentage of 34% or 31% inside and 21% or 13% outside depending if the game is indoors or outdoors.
Average number of field goal attempts needed to win shooting exclusively from inside or beyond the arc based on a set team average field goal percentage of 34% or 31% inside and 21% or 13% outside depending if the game is indoors or outdoors.

What is the ratio or distribution between shots taken from inside and beyond the arc?

Unless you play on a court where all field goals count as 1 point, you know that playing an varied game improves spacing and (theoretically) shooting efficiency. Based on the semi-informed estimate as to the FG% for FGAs inside and beyond the arc in the previous section, the next factor to estimate is FG distribution. To be sure, FG% and FG distribution fluctuate and co-influence each other between and within games but again I’ll attempt to base FG distribution on existing NBA and college level data, making educated guesses at how they might be adjusted.

League-wide averages for the NBA (2012-2013) report field goal attempt distributions at 75% inside the arc and 25% from outside the arc ( Finding a dearth of aggregated college data, I used the University of Texas at Austin’s 2012-2013 statistics for field goal distribution. Last year the Longhorns attempted 69% of their FGs from inside the arc and 31% from deep ( Let’s assume that the NBA-UT disparity is due to higher levels of organization and restraint at the pro level, meaning that we might decrement the pickup averages by the difference between the NBA and Texas numbers. This gives us a split of 63% from inside the arc and 37% from deep for indoor games. Outdoor averages for the NBA (80% and 20%) and the NCAA (77% and 23 %) are understandably more conservative so recalibrating the distribution for pickup puts us at 68% from inside the arc and 32% from beyond it (5% adjustment toward inside the arc is both triple the difference in discrepancy between pro and college outdoor distribution and a 5% bump up in the indoor rate dove-tailing with the 5% bump in pro-indoor and pro-outdoor distribution).

Estimated number of field goals needed to win at given scoring systems
Estimated number of field goals needed to win at given scoring systems

At those splits, based on the percentages calculated earlier, one tier and two outliers emerge. The main tier, including playing to 11 (both ways), 12, and 15 average around 40 FGAs per team per game, while playing to 21 by 2sN3s and 30 by 2sN3s come in well below (32.7) and above (45.4) respectively.

How might we estimate the effect of the And-1 rule on the number of FGAs needed to win?

I’ve only ever played on two courts were And-1 continuation was allowed. But, as one of those places is the University of Texas at Austin’s Gregory Gym (30 by 2sN3s) where I played pickup for seven years, I calculated the And-1 effect. When adding in an And-1 adjustment one must consider both the foul rate and the probability that the basket will go in despite the contact (or claimed contact) at each scoring interval. The most easily-available data on this dates back to the 2005-2006 NBA season ( that shows Shaq getting awarded foul shots on 8.5% of his FGAs. Meanwhile Ray Allen was given foul shots on 5.9% of his attempts that same year.

Shaq converting despite getting fouled.

Beyond a dirty look or two, calling a foul while in the act of shooting in an And-1 scoring system has no downside. If the shot goes in the shooter’s team gets the points, if the shot misses the mark, the shooter’s team gets the ball again. This dynamic, and the undisciplined nature of pickup defense, leads me to set the inside the arc called foul rate in And-1 scoring systems at 17%, double that of Shaq’s called foul rate (Shaq attempted no shots from beyond the arc in 2005-2006 ( Setting the foul rate for shots beyond the arc is a bit more difficult as no players listed shot solely from behind the arc. Ray Allen, still with Seattle at the time, took less than half of his FGAs from behind the arc ( and was probably fouled with greater frequency on his inside-the-arc FGAs than those he took from beyond the arc. I’m going to estimate/guess that he was fouled on 3% of his attempts from deep. In keeping with the formula above I’ll increment it to 6%.

Ray Allen, master of the four point play and corner three.

NBA players as a whole in 2005-2006 made the basket on shooting fouls 28% of the time ( I didn’t find data that broke this up by shot location so these calculations get extra flimsy. Based on lower overall pickup ball shooting percentages (and the fact that pickup ballers don’t practice shooting with contact) I’m going to reduce this number by 40% for shots inside the arc and by 70% for shots beyond the arc–giving us a basket-is-good conversion rate of 17% on shots inside the arc and 8% on shots from deep.

Based on those numbers, the FGAs to win on a court playing to 30 by 2sN3s allowing And-1s adjusts to 44.5 FGAs to win–thus reducing the number of FGAs to complete a game by about 1.

What is a reasonable turnovers per possession rate?

In making the jump from FGAs to possessions–possession being reset or incremented on every shot attempt–we’ve got to adjust for Ginobilis, I mean turnovers. If you’ve played enough pickup ball, you’ve witnessed or contributed to a game where the total number of turnovers (bad passes, stolen passes, traveling calls, double dribble calls, blocks…) exceeded the number of made FGs. I remember watching a game where more than two minutes went by without a FG attempt. As I mention in a post that develops a taxonomy of 25 different types of turnovers, empty possessions are natural within the course of game play, they are especially frequent when the players involved in a sequence are trying to do things above their capabilities.

During the 2013 Finals, the Spurs asked Ginobili to do more than he was able.

Since this feels like a time to be bearish on pickup ball, instead of taking the league averages I’ll pick teams performing near the bottom of their leagues. In the 2012-2013 season, the Houston Rockets turned the ball over on an NBA-worst 16% of their possessions ( while Mississippi State University basketball ranked at the bottom of the NCAA–giving back more than 24% of their possessions in 2012-2013 ( Incrementing the Bulldog’s figure by half the difference between it and the Rocket’s percentage gives us an average of one turnover per 3.5ish possessions or 28% of the time. Turnovers can be contagious and especially frequent in transition as players are in very different positions and moving at different speeds than they do when in half-court settings.

Transition play creates a-typical situations often resulting in turnovers.

What is the adjusted average number of total possessions to finish a game?

Based on these calculations of FG%, FG distribution, FGAs, and turnovers per possession the adjusted possessions for a single team to win is shown in the table below.

Adjusted number of possessions to win
Adjusted number of possessions to win

The only in-game event that creates separation between teams is offensive rebounds. In 2012-2013 the difference between the average offensive rebounding rate for the best offensive rebounding team in the NBA and the worst, was about 5 ( What’s more, the Spurs and Heat were in the bottom three in offensive rebounding rate while Milwaukee, Cleveland, Detroit, and New Orleans were in the top ten. According to calculations based on more than 20 years of NBA rebounding and possession data ( there is no significant correlation between offensive rebounds and winning unless you calculate shots per possession and winning percentage and even then it’s slight.

In my pickup basketball manifesto I mention how I used to go after offensive rebounds on a regular basis but felt that I could make more of a positive impact on my team by releasing right after a shot in an effort to get back on defense to limit fast-break and transition baskets. I pick my spots in terms of hitting the offensive glass now, only doing it when I trust my teammates to get back or when the feeling that we absolutely must score on a particular possession.

Since we’re counting an offensive rebound as a new possession even this weak correlation has no direct impact on our calculations (except for exposing how I’ve avoided calculating defensive and offensive rebounding rates). Given the lack of relevant correlations between winning and an advantage on the offensive boards, we’ll just multiply the per-team possession total by 2 to arrive at the per-game totals.

Estimated number of total possessions in a pickup game by scoring system.
Estimated number of total possessions in a pickup game by scoring system.

Part III: It’s about Time

What is the Average Possession and Game Length for a given game of pickup?

In 1954-1955 pro basketball’s governors instituted a shot clock ( The shot clock ensured that weaker teams or teams with a lead couldn’t just stall game play to increase their chances of winning. Having a set amount of time to shoot also rewards good defenses by forcing teams into bad shots near the end of the shot clock instead of letting them just continue to probe.

The average NBA game in 2012-2013 had 191 possessions ( and takes 48 minutes of timed play and about 2.5 hours of actual time to complete (yahoo message boards). After some digging and thinking about how one might calculate average possession length (APL) as well as average game length, the data seems useful even if the nature and culture of non-timed basketball games preempt concerns about stalling during pickup basketball.

Very few pickup players, much less teams, are willing to grind out a possession, patiently waiting for the best possible shot to come open via a pick and roll, off the ball screens, back cuts, post-re-post, or drive and kick, then reset and probe actions. Most pickup ballers want to push the ball up the court and attack as soon as possible (unless one of them happens to be trailing the play in which case he or she may call for everyone else to wait up). Most pickup ballers seem to be playing against an internal shot clock styled after the 2009 Phoenix Suns (if baseball would emulate the Suns’ pace of play it might actually be watchable).

Playing fast led to lots of easy baskets for the 2009 Suns.

The normal grammar of a possession in pickup ball goes something like these archetypal ‘plays’.

Rebounded Missed FG: Karl rebounds the ball. Karl passes to Larry who races up the court. Larry passes to a Chung-Kai who takes two dribbles and shoots a 5 footer.

  • Elapsed time: 3 seconds.

In-bounded after Made FG: Karl in-bounds the ball to Larry. Larry brings the ball across the half-court line and passes to Carlos who takes three dribbles and passes to Larry. Larry passes to Chung-Kai on the other wing who shoots a mid-range jumper.

  • Elapsed time: 11-15 seconds.

Offensive rebound: Karl gets the rebound two feet from the basket. Karl pump fakes twice and shoots from 2 feet away. Karl rebounds his own miss, pump fakes once and shoots from 2 feet away. Karl rebounds his own previously rebounded miss and shoots from 2 feet away.

  • Elapsed time: 5 seconds (3 possessions).

Backcourt turnover: Larry uses his freakishly long arms to poke the ball away from Team A’s point guard. Larry races down court and shoots a layup or pulls up for a shot from behind the arc while Team A’s point guard, too embarrassed to give chase, hangs his head and pouts.

  • Elapsed time: 3 seconds

Post-foul checked ball: Larry passes the ball to Carlos curling off a moving screen from Karl. Carlos immediately shoots an elbow jumper.

  • Elapsed time: 1 second

Defensive rebound: Raj grabs a rebound and throws the ball up-court ahead of what appears to be a sprinting Carlos. The ball sails over Carlos’ head and out of bounds.

  • Elapsed time: 2 seconds

After writing these out and thinking about personal experience. I wonder if the 2012 Clippers might be a better analogue than the 2009 Suns. These highlights (below) really capture the spirit of the pace and precision of pickup ball, plus Vinny del Negro has never looked better.

A more fitting analogue for fast-paced pickup ball.

In 2004, broke down NBA teams’ usage of the 24 second shot clock into four categories titled Quick (a FGA within 10 seconds), Early (11-15 seconds), Structured (16-20 seconds) and Late (21-24 seconds). While long possessions can happen at the end of tight games in pickup ball, the possession archetypes below suggest that most games experience a change or increment in possession by the 15 second mark, thus the re-calibration and category renaming.

Attack! A FGA (or turnover) in 1-8 seconds (6 second average). Most pickup fastbreaks take place within this time frame, also offensive rebound put-backs, and many checked-ball plays.

Thrust A FGA (or turnover) in 9-15 seconds (13 second average). Most half-court sets off of made baskets take about this long, also possessions executed by teams that have built up some familiarity and trust, and teams playing in their 3rd or 4th consecutive game.

911 Emergency A FGA (or turnover) in 16-20 seconds (18 second average). By this time in a possession, players either start to feel nervous, like the shot clock is about to go off and they should shoot to avoid ridicule, or they convince themselves that the reason none of the other players on their team have shot yet is because they want them to shoot. A contested fade-away or an over-dribbling-induced turnover are common outcomes.

Hot Potato A FGA (or turnover) in 21 or more seconds (25 second average). It’s the end of the game, the score is tight and the defense is pressing up on ball handlers (many of which don’t really have what could be considered a ‘handle’). What follows is a series of stalled dribble hand-offs up near half-court until a turnover, a called foul, or blind drive to the hoop transpires.

Of course there are teams that deliberately take 20 or more seconds per possession to let things develop. Such teams are usually older and have played pickup with each other for years or they played HS ball together for some single ‘A’ school in a city you’ve never heard of and ended up at the same university meaning they could keep the team together and dominate the pickup courts in an effort to build a legendary collection of RecSports IM Championship tee-shirts. While such teams are usually successful in pickup, they are rare.

Behold, the coveted Championship Tee-Shirt! Team photo of ‘Nashed Potatos and McGravy’ with their wearable IM basketball ‘trophy’ [photo:]
Despite the slightly different intervals, let’s use the 2004-2005 Suns shot clock usage distribution of 50%, 27%, 17%, and 6% ( to calculate the ATP for pickup basketball possessions for the respective Attack!, Thrust, 911 Emergency, and Hot Potato possession levels.

Estimated time (in minutes) needed to play each possession until a winner is declared
Estimated time (in minutes) needed to play each possession until a winner emerges

The final step in answering the question of time has to do with stoppage time. Not the futbol sense of stoppage time where they play extra time to make up for faked injuries and streakers. But rather the time it takes for teams to inbound the ball, track down an errant pass that has gone into the next court, or the time it takes when players argue over who scored last, which team is ahead, and whether or not the guy wearing the super-saggy shorts just executed a legal or illegal crab-step.

We’ll add 3 seconds for in-bounding the ball after made field goals. Five seconds for tracking the ball down after 25% of turnovers (the stoppage-of-play variety such as when ball soars out of bounds or someone calls a violation like traveling…), fifteen seconds per called foul–this gives time for some arguing about the legitimacy of the call as well as re-establishing the score. Finally, we’ll add 30 seconds per game for the inevitable impasse involving the only data points ballers care about. You know, the moment in the game when everyone has to count up how many times they scored and someone disagrees, or there’s some sort of Alpha Baller test of wills going on and neither side wants to back down.

Adjusted average time to finish a game of pickup
Adjusted average time to finish a game of pickup

Part IV: Scoring by 1s and 2s versus 2s and 3s

When I invited a few people who play a lot of pickup basketball to comment on an early version of this analysis I got the following response from Ben Shoemaker.

My biggest pet peeve with pickup [is] playing by 1’s and 2’s. It’s idiotic. Basically it just means that playing the right way and getting layups is dumb, and you should just jack up as many threes as possible. It is not basketball, it’s a game disguised as basketball

Ben’s perspective is shared by many and is borne out by the extrapolated analysis in this post when applied to indoor pickup ball. A post on the Basketbawful blog also points out the pitfalls of scoring by 1sN2s when playing indoors:

there are serious drawbacks to making the three-pointer worth double a normal basket. Players are much more likely to bomb away from downtown without regard to common sensibility. After all, hitting 25 percent of your twos is like hitting 50 percent of your ones, right? … This leads to many players shooting the three almost exclusively, and therefore not developing any other tangible on-court skill (like, say, passing the ball). It also creates circumstances under which a team can actually lose a game despite hitting almost twice as many field goals as the opposing team. Nothing is more frustrating than executing a precise and efficient offense and still losing to a bunch of selfish, one-trick gunners.

Recently, Ben Gaines wrote an extensive post about his issues with how scoring by 1s and 2s changes the game:

Let’s say I’m an NBA player who shoots 50% generally from inside the three-point arc and 40% outside of it. Some fans look at this and say, 40% * 10 three-point tries = 12 points and 50% * 10 two-point tries is = 10 points, so shouldn’t you always take the three? The answer is no, primarily because this faulty analysis ignores the fact that in organized basketball you are far more likely to get fouled and produce valuable free throws when shooting inside the three-point line, so your two-point tries are more valuable than they seem on the face of it. … you’re already more incentivized to play outside than you normally would be; why make things even worse by increasing the value of a three-pointer unnecessarily?

Personally, playing to 30 by 2s and 3s is my favorite scoring system. Games are long enough to have one or two swings in momentum and teams don’t overly benefit from shooting mostly long-range FGAs. In fact, according to the abbreviated table below, shooting mainly or only from inside the arc is more efficient than taking too many shots from deep.

Playing by 1s and 2s makes more sense on the playground where the absence of a bounded visual environment, an abundance of bright sunlight, and the presence of even a light breeze can combine to greatly reduce long-range shooting accuracy. The clip below shows relatively mild outdoor conditions and yet there are no jump shots made beyond 4′. Of course the players in the video are not exactly PUB-GOATs or Alpha Ballers but they resemble players I encounter all the time at the ARC on the UIUC campus. Attacks of poor shooting can happen in indoor pickup as well, but outdoor games just feel different. They don’t revolve around the three point line the way games tend to do indoors, especially when playing by 1s and 2s.

Outdoor conditions can wreak havoc on jump shooters.

My sense is that people score by 1s and 2s on indoor courts to feel closer to the street style of pickup ball, it connects them to playground basketball culture. The problem outlined by the Bens and Basketbawful is that shooting from behind the arc ends up being significantly more advantageous when playing indoors and thus has the potential to change the way the game is played. Based on extrapolated estimates of FG% (see table below), shooting exclusively from deep on an indoor court using the 15 by 1s and 2s scoring system makes for a 6 FGA advantage compared to shooting from inside the arc and a 4 FGA advantage when compared to the extrapolated FG distribution levels.

What scoring intervals tell us about shot distribution.
Games to 11 and 12 are typically outdoor scoring systems while playing to 15, 21, and 30 are usually indoor scoring systems.

The ARC gym at UIUC where I’ve played pickup basketball for the past 2 years at least allows the winning team to shift between the two systems, thereby offering teams the chance to negate the excessive advantage shooting from deep yields. Yet, when teams play 2s and 3s instead of 1s and 2s I haven’t noticed a tangible change in strategy based on scoring system. The game feels just as arc-centric when playing to 21 as it does when playing to 15. Still, based on the extrapolated data in this post, I’ve started to push the 21 by 2s and 3s whenever I can. If I regularly ran with a disciplined crew who understood the underlying advantages of playing the ‘right way’ it might have a bigger impact but when I play the role of primary ball handler I can influence the shots we take by looking to pass into the post more often. When I’m more of a role player on a team I try and adjust my own shot selection based on the scoring system. I’ve also worked to be more patient with chucker teammates when playing to 15 by 1s and 2s.

Part V: Coda

So what has this trek through different aspects of pickup ball given us? While most of the data used in the extrapolations have issues of validity and reliability, my goal was to offer a way of thinking through pickup basketball’s many scoring systems based on something more than (but still guided by) personal intuition. The result is a possible method or way forward for applying extrapolated data to the realm of pickup ball. Specifically it has added some strength to those who see scoring by 1s and 2s when playing indoors as problematic.

Overview of the estimated statistics for pickup basketball
Overview of the estimated statistics for pickup basketball (games to 11 and 12 played on outdoor courts, all others used for indoor pickup)

Like I said several thousand words ago, the journey through the world of extrapolated/estimated/invented pickup basketball statistics requires imagination, research, and observation. Hopefully you found it as entertaining to read as I found writing it to be. Also, consider yourself exempt from doing any further mathematics-related activities for the rest of the workday, maybe someday you can pay be back by picking me up when your team is running next, just tell the guy in the neon tank top that you forgot that you added me before you picked him up.

Jason Schwartzman’s math-class daydream.

4 thoughts on “The Algebra of Scoring and The Calculus of Keeping Score

  1. Pingback: Court Vision: Jim Buss says Dwight Howard ‘was never really a Laker’ | The Point Forward -

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