# Statistical Analysis Primer

Kevin Pelton, SUPERSONICS.COM | updated September 12, 2007

Statistical analysis of the NBA is still gaining attention, both inside the league and from fans and analysts. For newcomers, SUPERSONICS.COM has put together a quick primer explaining the key statistics, thinking and names you need to know.

Possessions - Arguably the most important discovery made by statistical analysts in basketball is the critical importance of possessions. If a possession is considered as ending with a made shot, a defensive rebound or a turnover - that is, an offensive rebound is not a new possession - the two teams in any given games are essentially limited to the same number of possessions, other than the possibility of getting one extra possession in each half. Because of this, being more efficient in a game with your possessions means you will almost certainly win.

Where per-possession statistics are particularly valuable is in comparing teams that play at different paces. Pace is statistically defined as the number of possessions per game or per 48 minutes (to account for overtimes).

While possessions can be tracked, this is not done officially, meaning they are usually estimated from other team statistics. This is done using the following formula:

Pos = .96 * (FGA + (.44*FTA) - OR + TO)

The .44 multiplier is because not all free throws take up a possession. Technical foul shots and "and-ones" do not, while there are more than two free throws on one possession with a three-shot foul. Research has determined that about 44% of all free throws take up possessions, thus .44 is used as the multiplier. The .96 multiplier accounts for team offensive rebounds in situations where a missed shot is tipped out of bounds by a defensive player, continuing the possession without an offensive rebound being credited.

The high-octane Phoenix Suns were passed up in pace by a pair of teams in 2006-07. The Golden State Warriors averaged a league-high 97.6 possessions per 48 minutes, followed by Denver (96.2) and the Suns (94.2). Detroit (86.1) played the NBA's slowest pace, while the Sonics were almost exactly average at 90.6.

Offensive Rating = Pts/Pos * 100Defensive Rating = OppPts/OppPos * 100

Traditionally, per-possession team ratings - the best method of evaluating offensive and defensive performance - have been presented on a per 100 possessions basis.

The Sonics ranked 12th in the NBA in Offensive Rating, averaging 108.4 points per 100 possessions. Phoenix (116.1) led the league. On defense, the Sonics allowed 112.0 points per 100 possessions to rank 27th in the NBA. The Chicago Bulls (101.1) edged out San Antonio for the top spot.

Per-Minute Statistics - Another important breakthrough for analysis of the NBA was finding that statistics calculated on a per-minute basis tend to be fairly consistent even when a player changes his role and begins to play more minutes. This allows for a level playing field in comparisons of low-minute reserves (as long as they've played a reasonable number of minutes; most cut-offs are 500 or 1,000 minutes for the season) and starters. Numbers are usually expressed on a per-40 minute basis, but also occasionally per 48 minutes. Sometimes, you'll hear this referred to as a player's rate; his "scoring rate", for example, would be points per 40 minutes.

The use by analysts of per-minute statistics allowed players like Andrei Kirilenko, Michael Redd and Zach Randolph to be identified as future stars even when they were languishing on their team's bench.

Stat/Min * 40

Rebound Percentage - While rebounds per 40 minutes improves upon traditional methods for judging rebounders, this skill can better be evaluated by taking into account that some players have the opportunity to grab more rebounds than others. The most fair way to evaluate rebounders is by percentage of all missed shots when they are in the game that they rebound, typically estimated by their team's and opponent's rebounds per minute. This is known as Rebound Percentage (Reb%) or, occasionally, Rebound Rate.

Reb% = Reb / (((TmReb + OppReb)/TmMin)*Min)

Center Nick Collison led the Sonics in 2006-07 by grabbing 16.8 percent of available rebounds. One-time Sonics forward Reggie Evans led the league in rebound percentage for the third straight season, pulling down 23.0% of available rebounds.

At the team level, rebound percentage takes into account the fact that good teams usually outrebound their opponents because defensive rebounds are easier to get than offensive rebounds. The total team rebounding percentage is the average of its offensive and defensive rebounding percentages.

Team OReb% = TmOReb / (TmOReb + OppDReb)Team DReb% = TmDReb/ (TmDReb + OppORebTeam Reb% = (Team OReb% + Team DReb%)/2

Player rebounding percentage can also be split into offensive and defensive rebounding, which can prove insightful because few players are equally adept at both. At the team level, there is actually surprisingly little relationship between offensive and defensive rebounding, probably because offensive rebounding depends heavily on whether the coach chooses to crash the boards or play back to prevent fast breaks.

For example, the Houston Rockets were the NBA's best team on the defensive glass (77.0%) but ranked 22nd in offensive rebounding percentage (25.7%), indicating Jeff Van Gundy's bias toward defense. As for the Sonics, they ranked 14th in offensive rebounding (27.8%) but just 27th on the defensive glass (70.9%).

Shooting Efficiency - If there is an on-base percentage in the NBA - a statistic that has traditionally been undervalued - it would probably be some measure of a player's efficiency in scoring points. There's a stereotype that all statistical analysts think Allen Iverson is a bad player due to his low shooting percentage that is untrue because Iverson's ability to create shots and get his teammates better looks is valuable. Still, being efficient with your shots is very important. The two most common ways of measuring the concept of shooting efficiency are Effective Field-Goal Percentage (eFG%) and what this site calls True Shooting Percentage (TS%).

Effective Field-Goal Percentage was popularized by current L.A. Clippers Coach Mike Dunleavy and the Rick Barry's Pro Basketball Bible series. It adjusts for the added value of three-pointers by counting them as 1.5 field goals, thus make it more fair to three-point shooters than field-goal percentage.

eFG% = (FGM + .5*3PM)/FGA

True Shooting Percentage goes a step further by factoring in a player's performance at the free-throw line and considering their efficiency on all types of shots.

TS% = Pts/(2*(FGA + (.44*FTA)))

Former Sonics guard Brent Barry - Rick's son - led the NBA in both categories in 2006-07, posting a 62.6% effective field-goal percentage and a 66.6% True Shooting Percentage. Barry has led the NBA in True Shooting Percentage three times, including twice while in Seattle. Rashard Lewis (58.7%) was the most efficient Sonics shooter by True Shooting Percentage.

Linear weights - The most common way of evaluating players' overall ability is through the use of what's known as "linear weights" formulas, so named because they assign a weight to each statistic (rebounds, steals, points, etc.) and add or subtract them. The most commonly used linear weights are the NBA.com Efficiency System, David Berri's Wins Produced and John Hollinger's Player Efficiency Rating (PER). PER is the most familiar to fans and the most likely to be used on this site. 15 is average for Hollinger's PER, bigger numbers better and smaller worse.

Ray Allen's 21.8 PER led the Sonics in 2006-07. Dwyane Wade (29.2) topped the NBA.

Plus-minus statistics - The current area of growth in statistical analysis is in the field of plus-minus statistics. At its most basic level, plus-minus merely evaluates how well a player's team plays when he's on the court. This has been taken a step further by taking it on a per-48 minutes or per-100 possession basis and comparing it to how the team does without a player. The difference in these two was formerly known as a player's "Roland Rating", after Roland Beech of 82games.com and now is known as net plus-minus.

Amongst Sonics regulars, Lewis (+5.7) had the best net plus-minus rating in 2006-07.

The next step is adjusting for the strength of a player's teammates and the opponents he faces. This is known as adjusted plus-minus. The concept was pioneered by Indiana University professor Wayne Winston and statistician Jeff Sagarin (more famous for his college football and basketball ratings), whose WINVAL program is used by the Dallas Mavericks and was used by the Sonics. University of North Carolina-Greensboro professor Dan Rosenbaum has extended Sagarin's and Winston's work by combining adjusted plus-minus data with traditional statistics in his DanVAL system.

Pythagorean Record - At the team level, analysts prefer to use measures based on a team's point differential rather than actual win-loss record because wins in close games tend not to reflect a team's true skill. Teams that start the season with a better record than their point differential tend to slow down and vice versa. Point differential is also a better predictor of future performance than win-loss record.

The actual "Pythagorean" method for calculating expected record is borrowed from baseball, where it was pioneered by Bill James. It takes the general form PF^X / PF^X + PA^X, where X is an exponent that varies depending on the sport. In the NBA, the preferred exponent is 16.5. This site generally uses Expected Wins, found by the relationship that each point per game of differential is on average worth 2.7 wins over the course of the season:

ExWins = 2.7 * Diff + 41 ExWin% = 2.7 * Diff + 41 / 82

Web sites - Naturally, the growth of the Internet has helped spur statistical analysis of the NBA. Here are some key sites worth visiting: