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# Statistical Analysis Primer

Statistical analysis of basketball is still gaining attention, both from NBA and WNBA teams and from fans and analysts. For newcomers, stormbasketball.com has put together a quick primer explaining the key statistics and thinking.

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 40 minutes (to account for overtimes).

Possessions have actually been tracked on a game-by-game basis in the WNBA since 2004, but for comparison purposes we estimate possessions 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.

Defensive Rating - points allowed per 100 possessions - uses precisely the same theory, but at the defensive end of the court.

Per-Minute Statistics - Another important breakthrough for basketball analysts was finding that statistics calculated on a per-minute basis tend to be fairly consistent even when a player changes her 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 250 or 500 minutes for the season) and starters. Sometimes, you'll hear this referred to as a player's rate; her "scoring rate", for example, would be points per minute. Traditionally, this rate is multiplied by the 40 minutes in a game so the numbers make more sense.

Stat/Min * 40

Rebound Percentage - While rebounds per 40 minutes is a very good way to evaluate rebounders, you can do even better 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. (This is usually estimated by their team's and opponent's rebounds per minute). This is known as Rebound Rate or Rebound Percentage (Reb%).

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

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 + OppOReb
Team 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.

Shooting Efficiency - If there is an on-base percentage in basketball - 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 a low-percentage shooter like Betty Lennox is a bad player that is untrue because Lennox'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 True Shooting Percentage (TS%).

Effective Field-Goal Percentage was developed by former L.A. Clippers Coach Mike Dunleavy and popularized by 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 3-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)))

Four Factors - The Four Factors were designed by pioneering APBRmetrician Oliver to measure the critical parts of team performance - shooting (eFG%), getting to the free-throw line (FTA/FGA), rebounding (OR%/DR%) and avoiding turnovers (TO%). Naturally, these same factors also apply at the defensive end, so there are in a sense really eight factors. Of these, shooting is the most important for a team to control, and about 40 percent of a team's performance is explained by its shooting and that of its opponents. Turnovers (25%) are second in importance, followed by rebounding (20%) and free throws (15%). Using the Four Factors and team rankings, it is easy to spotlight areas of improvement or need for a team.

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 WNBA.com Efficiency System and John Hollinger's Player Efficiency Rating (PER). 15 is average for Hollinger's PER, bigger numbers better and smaller worse - except when rating the performance of a player defensively, when you want the opponents to have smaller numbers.

WARP - Exclusive to stormbasketball.com, this rating system is built on the theory of creating an imaginary team of four average teammates with the player in question. Using player and team statistics, we calculate the Offensive and Defensive Ratings of this "team," which are used to estimate its winning percentage. Players create value by producing better ratings than a team with four average players and one at replacement level (the level of freely available talent, including players invited to training camp). WARP stands for Wins Above Replacement Player, which is based on the per-minute Win% and minutes played.

Plus-minus statistics - At its most basic level, plus-minus merely evaluates how well a player's team plays when she is on the court. Plus-minus statistics for individual games are now available on boxscores at WNBA.com. This has been taken a step further by taking it on a per-40 minutes or per-100 possession basis and comparing it to how the team does without a player to isolate their impact. The difference in these two is known as net plus-minus.