Rankings (mathematics)
Rankings in mathematics refer to the process of ordering objects or entities based on specific criteria, a practice that has been part of human history for centuries. This concept applies across various domains, including sports, education, and social sciences, where rankings are often employed to assess performance or quality. For instance, high school valedictorians are determined by their academic standings, while institutions like US News and World Report rank colleges using a mixture of quantitative and qualitative indicators.
Mathematical techniques utilized to study rankings encompass algebra, geometry, graph theory, and game theory, among others. Furthermore, nonparametric statistical methods allow for the analysis of data that do not adhere to traditional assumptions, expanding the application of rankings in diverse fields. Interestingly, the interpretation of rankings can vary; for example, while it is preferable to hold a higher rank in many scenarios, percentile rankings present a different perspective, where higher values indicate better performance relative to peers. In addition to academic and sports contexts, the term "rank" has specific mathematical meanings, such as the rank of a matrix or in graph theory, demonstrating its multifaceted nature. Understanding rankings requires careful consideration of the criteria and methods used, as they can significantly influence the conclusions drawn from the ordered data.
Rankings (mathematics)
Summary: Ranking is a widely used to create ordered lists of people or objects, and there are many ways to assign and analyze ranks.
Throughout human history, people have been ordering objects into hierarchies based on criteria such as measurements or qualitative properties. In the twenty-first century, people rank many objects, such as quarterbacks, political candidates, and restaurants. Every spring, high school seniors eagerly wait to see who will be the valedictorian, or top-ranked student, of their high school class. However, there is not usually a single unique ranking for a set of objects, since ranks depend on the criteria selected and the specific method in which they are combined. US News and World Report aggregates multiple quantitative and qualitative indicators in its annual ranking of colleges. Mathematicians use a variety of techniques to study ranking, such as algebra, geometry, graph theory, game theory, operations research, and numerical methods. An entire subset of statistical techniques based on ranks, called nonparametric or distribution-free tests, are used to transform and analyze data that do not conform to the assumptions or parametric tests.
These techniques are often used in the social sciences. There are also debates about whether ranks are true numbers, given that the spacing between ranks need not be equal in the manner of most common measurement scales. For example, the difference between one inch and two inches is the same as between two inches and three inches. The difference between first and second place, however, is not necessarily quantitatively or qualitatively the same as the difference between second and third place.
Sports
Athletic competitions are one very visible use of rankings. During the ancient Olympic Games, athletes would compete in events, such as running, boxing, and the pentathlon, to determine which athletes were better than others. Ultimately, they would be ranked by their performance in these events. Even during the modern Olympics, though the events are more numerous and athletes generally compete in only a few events, the result is a ranking of the best athletes, with prizes being awarded to the top three finishers. There are rankings for other sports as well. For example, the Associated Press ranks the top 25 NCAA football teams by polling sportswriters across the nation. Each writer creates a personal, subjective list of the top 25 teams from all eligible teams (more than 25). The individual rankings are then combined to produce the national ranking by giving a team 25 points for a first place vote, 24 points for a second place vote, and so on down to one point for a 25th place vote. Teams are also regularly ranked by their number of wins or other game-related metrics, as are individual players.
Tests
Rankings also occur on standardized tests. Rather than give each individual a unique rank, tests such as the SAT separate the scores into percentages and then rank test takers according to the percentage they fall into. Percentile ranks can also be seen in other places, such as height and weight charts for children. Whereas many rankings place an emphasis on small numbers (it is better to be ranked first or second than twenty-fifth), percentiles are considered in the opposite manner—a larger value percentile ranking is a better rank. Percentiles indicate what percentage of the test-taking group performed the same or worse than a test-taker in that percentile. For example, being in the 57th percentile would indicate that 57 percent of the test takers scored the same or worse. When considering rankings, it is important to determine how the ranking is arranged to properly interpret the data.
Other Mathematical Connections
The word “rank” carries many specific definitions in various fields of mathematics. For example, the rank of a matrix is the number of linearly independent rows or columns. In graph theory, the rank of a graph is the number of vertices minus the number of connected components. Other definitions of rank can be found in set theory and Lie algebra (named for mathematician Sophus Lie). In chess, a game studied by many mathematicians, a rank is a row on the chessboard.
Bibliography
Gupta, Shanti, and S. Panchapakesan. Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations. Philadelphia: Society for Industrial Mathematics, 2002.
Marden, John I. Analyzing and Modeling Rank Data. New York: Chapman & Hall, 1995.
Winston, Wayne. Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football. Princeton, NJ: Princeton University Press, 2009.