Basketball statistics

Summary: Play can be analyzed geometrically and probabilistically to inform strategy or construct simulations.

Basketball is an international sport that can be enjoyed either as a participant or as a spectator, regardless of one’s sex or one’s age. A growing number of coaches, reporters, and ardent fans are using mathematics to examine all aspects of basketball—the physical aspects and performance of its players, the analysis of each element (shooting, defense, strategies) of the game, and the combined geometry and physics surrounding the game. Perhaps as expected, this mathematical analysis can have opposite effects, either enriching or ruining the sports experience itself.

Basketball was intended to be a dynamic, fair competition between two teams; however, mathematical concepts and techniques can be used in a basketball environment to identify patterns of strengths and weaknesses, suggest optimal strategies for coaches and players, stimulate discussions, and resolve arguments. Statistician Dean Oliver is a well-known contributor to the statistical evaluation of basketball, which is called APBRmetrics. The name comes in part from the Association for Professional Basketball Research (APBR). This methodology is a very similar to the analysis of professional baseball using sabermetrics. Though difficult to implement practically, geometry, trigonometry, and calculus can shed light on these important ideas:

• Given a player’s height, the best angle and velocity for shooting a basketball, assuming the intent is to have the basketball’s parabolic arc pass through the basket (often called the “Shaq phenomena”)
• The connection between the angle of shooting a ball and the event known as an “all-net” basket
• The connection between a player’s height and where a player should aim a shot—at the center of basket, the front of the rim, or the back of the rim
• Use of angles in making bounce passes
• Determining defensive positions that maximize centers of gravity
• The connection between a player’s position on the court and decisions to bank the basketball off the backboard as the best shot
• Comparison of the merits of shooting a free-throw underhand versus overhand

By gathering and analyzing the myriad of available data provided by a game experience, mathematical probabilities can help examine the chances of particular events happening within a game, including the following:

• The likelihood of a player making 0, 1, or 2 points in a 1-and-1 free throw opportunity
• The reality of a player having a “hot-hand,” based on his or her making successive shots
• The decision as to which player should be purposely fouled at the end of a close game
• The evaluation of a player’s performance in terms of “per-possession efficiency”
• The probability of a record being broken, either by a team or a player

Similarly, the collection and organization of mathematical statistics can provide perspectives that explain game occurrences, provide comparative rankings of teams and players, and assist in future decision making by coaches and team management. The usual sources of statistics are data regarding shooting, rebounding, free throws, turnovers, defensive gains, and time management. Some specific examples include the following:

• The simple use of ratios, means, and medians as descriptive statistics for a player, a position, a game, or a season
• Connections between a player’s characteristics and training regimens relative to game performance
• Trend analysis, based on either a player’s or a team’s performance in specific ways over the past five, 10, and 15 games
• Winning tendencies based on connections to lead changes during a game or knowledge of the team leading at the end of the third quarter
• The impact of rules changes on scoring and defenses within the sport itself, such as the observed effects of expanding either the three-point arc or the free-throw lane
• Determining the “best” all-time player in a particular position (for example, center), at a particular time in a game (for example, last-second shot), or in an era
• The seeding and selection of teams in a bracketed tournament, possibly as part of a betting pool with stated odds
• Selection of players by professional teams during the annual draft, using historical data for each player’s performance in conjunction with physical data
• The use of statistical data as part of contract negotiation between players and management
• The release or trading of players based on team needs

The ideas of mathematical game theory have been applied to the decision-making process within a basketball environment, leading to choices of optimal tactics. The specific decisions range considerably:

• A coach’s choice of designed offenses and defense strategies, relative to the opposing coach’s choices
• A coach’s calling of time-outs at opportune times within a game
• A coach trying to influence or reverse decisions by game officials
• A coach’s use of techniques to motivate specific players
• A team’s selection of players during a draft, dependent on the player’s apparent abilities, the inferred needs of other teams, and the specific draft round
• Contract negotiations involving players, agents, and team management

Finally, using all of these available statistical data and mathematical modeling techniques, one can create realistic simulations of basketball events, full games, or even tournament series. At the collegiate and professional levels, coaches are increasingly using mathematics to remain competitive, even hiring mathematical statisticians as important parts of their staffs. Some mathematicians are even found on the court.

Retired San Antonio Spurs player Michael Robinson earned a bachelor’s degree in mathematics from the U.S. Naval Academy, and is considered by many to be the best basketball player that school has ever seen. However, there are still some authors and fans who suggest the team with the best players and coaches will usually win, despite the use of sophisticated mathematics.

Bibliography

Bennett, Jay, and James Cochran. Anthology of Statistics in Sports. Philadelphia: Society for Industrial and Applied Mathematics, 2005

De Mestre, Neville. The Mathematics of Projectiles in Sport. Cambridge, England: Cambridge University Press, 1990.

Friedman, Arthur. The World of Sports Statistics: How the Fans and Professionals Record, Compile, and Use Information. New York: Athenaeum, 1978.

Oliver, Dean. Basketball on Paper: Rules and Tools for Performance Analysis. Washington, DC: Brassey’s, 2004.