*This post originally appeared at Hickory-High.com on 11/06/2013*

The value of the three-pointer is a staple of basketball analytics. It is often referred to as 'advanced,' but really it is mostly the result of pretty simple algebra, a team has to hit 50% more two point shots to get the same number of points as they would get off of three point shots. Away from the rim, most players struggle to do that.

Of course, there is more to it than that, all kinds of nuances can effect the percentages of any shot. Defenses can do the same math and prioritize guarding the three point line. When misses do happen there are different expected percentages of offensive rebounds. It is easier to for the opposing team to score off of defensive rebounds, which add up with more misses from distance. Driving to the rim results in more free throws, which is a very efficient way to score.

Then there is the issue of 'spacing,' simply for its own sake. Kirk Goldsberry, in a Sloan paper, estimated the scoring area in basketball is 1280 square feet based on where the vast majority of shots are taken. From a geometry stand point taking shots from further out increases the percent of that area the five defenders have cover and increases the space between defenders, opening lanes to the basket. As Houston GM Darryl Morey or any old time basketball hand would probably tell you, driving to the rim and spacing are complimentary. Players need space to open lanes and drive to kick it back out.

A while ago I ran a number of K-means Cluster analyses based on shot type using data from the suddenly defunct, HoopData. The analysis grouped Guards, Wings and Bigs based on where they shoot in relation to the basket. The analysis grouped both the guards and the wings into groups of Stretch and Slasher types, scoring from the three point line and at the rim respectively. Meanwhile, Bigs fell into three clusters, Bigs that shot almost exclusively at the rim, those with more varied offensive games and stretch bigs, who added a three point shot.

For example, Stretch Wings averaged 56% of their shots from three point range and 19% at the rim, while Slashing Wings averaged 21% from three and 35% at the rim, according to my analysis. The numbers for Stretch and Slashing guards were both very similar. The cluster analysis allows me to group player types together to look for patterns in the data, for example, I found some interesting patterns in terms of style of play with Stretch back court players and wings have higher eFG% percentages, but Slashers get more offensive rebounds and more free throws. Again, I will use the Wings as my example.

Group | COUNT | Avg TS% | Avg eFG% | Avg ORB% | Avg DRB% | Avg AST% | Avg TOV% |

Slash Wing | 39 | 52.7% | 49.1% | 5.84 | 16.09 | 10.79 | 11.95 |

Stretch Wing | 25 | 53.7% | 51.2% | 3.20 | 13.08 | 10.03 | 11.15 |

Grand Total | 64 | 53.1% | 49.9% | 4.81 | 14.91 | 10.50 | 11.64 |

But to take the analysis to the next step I wanted to look for an independent value of spacing, and its limits. I think that is another value of using categorization, Stretch players are measured as players that take more three point shots, not necessarily make them at a higher rate (though there is a correlation there too). The methodology adopted was both to chart the Stretch players in five man line ups against offensive rating (ORTG) and, secondly, to calculate an expected ORTG, (points per one hundred possessions) based on the individuals in a line up and compare it to the actual ORTG generated by that five-man lineup, then add Spacing configurations to the mix. (Similar to an analysis done here by Jacob Frankel using a different definition of stretch.)

I used line up data from the last two years, courtesy of NBA.com's media stats site. The chart below uses a cut off of five-man lineups that played at least 250 minutes together in the either of the last two years, however, I ran the same chart multiple times with different minute filters and got largely the same results. With the five man line up at this cut off, **Tyson Chandler**, **Carmelo Anthony**, **Jason Kidd**, **Raymond Felton** and **J.R. Smith** had the highest ORTG with 119.4 points per 100 possessions, and **Brandon Knight**, **Jason Maxiell**, **Greg Monroe**, and **Rodney Stuckey** had the lowest at 93.6 points per 100 possession.

I used a polynomial line to visualize the trend. Consistently, with different minute filters the trend showed an increase in offensive rating going from 0 Stretch players in a line up to 1 and 2, and then flattened. The same pattern was seen with Wing and Guard Stretch players only, as well as, using effective field goal percentage. Then I ran the expected ORTG for each line up against the actual ORTG the line up had, with the 250 plus minute filter the R^2 was a decent 0.434. The chart below shows the number of Stretch shooters in each line up against the residual error on the expected ORTG.

Again the same pattern can be seen with an increase in the offensive rating with two stretch players on the court and a then leveling off indicating diminishing returns to spacing. Using that information I made a 'dummy' variable for each lineup that had two or more Stretch shooters and ran that variable in a step wise regression along with the Expected ORTG and linear count of Stretch shooters. In every minute filter the Dummy Stretch came out as the most significant contribution in addition to the Expected ORTG. The result of the model run with the 250 minute filter indicated that **the value of having at least two Stretch shooters on the floor is 2.44 points per hundred possessions**.

Unstandardized Coefficients | Sig. | 95.0% Confidence Interval for B | |||

Variable | B | Std. Error | Lower Bound | Upper Bound | |

(Constant) | 13.29 | 12.53 | 0.29 | -11.74 | 38.31 |

Expected ORTG | 0.83 | 0.12 | 0.00 | 0.60 | 1.07 |

StrechersDummy | 2.44 |
0.89 | 0.01 | 0.67 | 4.21 |

The results were similar with other filters on minutes, with the independent value of having two Stretch shooters in a line up centered just above two points per hundred possessions. It might not seem like a lot, however, this value represents the value above the simple algebraic shooting percentage multiplied by three points. And two net point per game equal about 5 wins over the course of the season.

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