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when the Baltimore Ravens announced in August that they had hired a director of football analytics, it was a rare public signal of the growing interest among teams in weaving statistical analysis into game-day, draft and free-agency preparation, and even into the management of workouts and injury rehabilitation

With advanced statistics, he notes, teams are able to see trends and adjust in real time. It used to be that teams would look back at how often they ran a play and how much it gained. Now, do they want to know Cam Newton’s completion percentage when a defense rushes three? Or four? Or six or more? That information is available week to week, allowing teams to tailor game plans with far greater specificity. Much of the work is also centered on figuring out some of the game’s most vexing problems — when to kick a field goal versus going for it on fourth down; what to do under the new overtime rules; when to challenge a call; when to use a timeout — amid the chaos of the sideline.

For most teams, though, the most intriguing application may come in player evaluation — projecting how college players will perform in the N.F.L. and figuring out how valuable one player compared with another

Powerful statistical algorithms allow us to connect these patterns to individual learning success.

After decades of urban evolution, the world’s major subway systems appear to be converging on an ideal form. On the surface, these core-and-branch systems — evident in New York City, Tokyo, London or most any large metropolitan subway — may seem intuitively optimal. But in the absence of top-down central planning, their movement over decades toward a common mathematical space may hint at universal principles of human self-organization. Understand those principles, and one might “make urbanism a quantitative science, and understand with data and numbers the construction of a city,” said statistical physicist Marc Barthelemy of France’s National Center for Scientific Research.

On the surface, these core-and-branch systems — evident in New York City, Tokyo, London or most any large metropolitan subway — may seem intuitively optimal. But in the absence of top-down central planning, their movement over decades toward a common mathematical space may hint at universal principles of human self-organization.

With equations used to study two-dimensional spatial networks, the class of network to which subways belong, the researchers turned stations and lines to a mathematics of nodes and branches. They repeated their analyses with data from each decade of a subway system’s history, and looked for underlying trends. Patterns emerged: The core-and-branch topology, of course, and patterns more fine-grained. Roughly half the stations in any subway will be found on its outer branches rather than the core. The distance from a city’s center to its farthest terminus station is twice the diameter of the subway system’s core. This happens again and again.

Nor is it clear that the math we learn in the classroom has any relation to the quantitative reasoning we need on the job. John P. Smith III, an educational psychologist at Michigan State University who has studied math education, has found that “mathematical reasoning in workplaces differs markedly from the algorithms taught in school.”

It’s not hard to understand why Caltech and M.I.T. want everyone to be proficient in mathematics. But it’s not easy to see why potential poets and philosophers face a lofty mathematics bar. Demanding algebra across the board actually skews a student body, not necessarily for the better.

Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives. Thus mathematics teachers at every level could create exciting courses in what I call “citizen statistics.”

The impact of data abundance extends well beyond business. Justin Grimmer, for example, is one of the new breed of political scientists. A 28-year-old assistant professor at Stanford, he combined math with political science in his undergraduate and graduate studies, seeing “an opportunity because the discipline is becoming increasingly data-intensive.” His research involves the computer-automated analysis of blog postings, Congressional speeches and press releases, and news articles, looking for insights into how political ideas spread.

But the computer tools for gleaning knowledge and insights from the Internet era’s vast trove of unstructured data are fast gaining ground. At the forefront are the rapidly advancing techniques of artificial intelligence like natural-language processing, pattern recognition and machine learning.

Dr. Atkinson, an expert at applying mathematical methods to linguistics, has found a simple but striking pattern in some 500 languages spoken throughout the world: A language area uses fewer phonemes the farther that early humans had to travel from Africa to reach it.

Dr. Atkinson is one of several biologists who have started applying to historical linguistics the sophisticated statistical methods developed for constructing genetic trees based on DNA sequences.