Football statistics

Summary: Football coaches use statistics to inform their decisions while the National Football League analyzes the effects of its rules.

Though a physical battle between two teams of talented athletes, football can be analyzed using mathematical ideas and techniques. Pertinent to coaches, players, fans, and betting agents, these analyses focus on all aspects of football—the physical aspects and performance of players, game elements (passing, running, defense, and kicking, as well as strategies), and the geometry-based physics surrounding the game. Mathematical analysis can impact the game positively or negatively. Nonetheless, football remains a physical competition between two teams, despite the use of mathematics to identify patterns of strengths and weaknesses, suggest optimal strategies, provide rankings, stimulate discussions, and possibly resolve arguments.

Quarterback Rating

The National Football League uses a mathematical formula to rate quarterbacks. Data are collected for each game and for the season relative to a quarterback’s pass completion percentage (P), touchdown pass percentage (T), pass interception percentage (I), and average gain per attempt (G). Using a few boundary conditions, a quarterback’s rating (Q) is determined by the formula

The formula’s derivation in terms of four independent variables involves multiple regression techniques.

Overtime Rules

The National Football League also uses Markov chain techniques to analyze its overtime rules in response to the “statistical fact” that too many football teams were winning important games with a field goal on their first overtime possession. Thus, the “winning” team, after a hard-fought game, is influenced too greatly by a single coin flip that determines team possession, with minimal differences accounted for by a team’s ability to score on the first possession. Effective in 2011, the rules for play-off games were changed to prevent the game ending with a field goal on the first possession of overtime.

Though difficult to implement practically, geometry, trigonometry , and calculus all play strong roles within a football game and its situations. Examples include the following:

• • Use of a quarterback’s physical characteristics to determine the best angle and release points for throwing a pass, assuming it must reach receivers in different field locations and at multiple distances
• • Use of the law of cosines to both understand and improve passing angles, timing, and patterns run by receivers
• • Determination of an optimal efficiency for punters on each kick, or the ratio of the actual kick’s distance to the maximum possible distance using the same force
• • Prior to kicking a field goal, determination of success in terms of the angle subtended by the two goal posts
• • Determinination of a defensive lineman’s stance to maximize centers of gravity and potential force on impact with an opposing linemen

By gathering and analyzing the available data provided by a game, probabilities can help examine the particular events happening within a game, such as the following:

• • Likelihood of a team making 0, 1, or 3 points after a touchdown score
• • Reality of a quarterback having a “hot hand” in his or her completion of successive passes
• • Success of making a field goal, given it will or will not result in a change in who has the leading score
• • Probability of scoring during a fourth-and-goal
• • Monitoring a coach’s decisions in calling plays, especially if “conservative”
• • Probability of a record being broken, either by a team or by a player

Similarly, mathematical statistics provide perspectives that explain game occurrences, provide comparative rankings of teams and players, and assist in decision making by coaches and team management. The usual sources of statistics are data regarding passing, running, defense, kicking, turnovers, and time management. Examples include the following:

• • Use of ratios, means, and medians as descriptive statistics for a player, a position, a game, or a season
• • Use of logistic regression models to calculate end-of-game point differentials, based on independent variables such as turnovers, passing yardage, running yardage, penalty yardage, number of first downs, and number of completed passes
• • Impact of icing a place-kicker at crucial times within a game
• • Correlations 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 5, 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
• • Impact of rules changes on team scoring and defenses within the sport itself, such as observed effects of initial field positions subject to penalties or punts out of bounds
• • Determining the “best” all-time player in a particular position (for example, quarterback, tight end, halfback, linebacker, or field-goal kicker), at a particular time in a game (such as the last quarter) or in an era
• • The use of digraphs and “mysterious” statistical formulas to determine weekly rankings and placement of teams in a bracketed tournament (such as the Bowl Championship Series), directly affecting betting pools 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 negotiations between players and management, or even the release or trading of players based on team needs
• • The questionable yet significant correlation between stock market performance and the Super Bowl’s winning team

Mathematical game theory is evident in a coach’s decision-making process, such as on each play within a football game, hoping to choose optimal tactics. The specific decisions range considerably and include the following:

• • A coach’s choice of designed offensive plays and defensive set-ups, relative to the down, position on the field, time of game, score, and opponent
• • A coach’s calling of time-outs and play reviews at opportune times
• • A coach’s use of techniques to motivate specific players
• • A team’s selection of players during a draft, dependent on the players’ apparent abilities, the inferred needs of other teams, and the specific draft round
• • Contract negotiations involving players, agents, and team management

Finally, using these statistical data and mathematical modeling techniques, one can create realistic simulations of football games, possibly using computer animations.

At the collegiate and professional levels, coaches increasingly use mathematics to remain competitive, even hiring mathematical statisticians as important parts of their staff. However, some authors and fans suggest that the football team with the best players and coaching will usually win, despite any use of sophisticated mathematics.

Bibliography

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

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

Eastway, Rob, and John Haigh. Beating the Odds: The Hidden Mathematics of Sport. London: Robson Books, 2007.

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

Gay, Timothy. The Physics of Football. New York: HarperCollins, 2005.