Five years later, knowing that his background in finance might be seen as an asset in baseball, Weber reached out to Yankees General Manager Brian Cashman. Two weeks before the start of spring training in 2009, Cashman offered him a job.

“When I was hired initially, I wanted to make sure it would be O.K. if I worked in the off-season,” Weber said. “I’m very interested in understanding the analytics side of baseball — it certainly piqued my interest in how the game is changing.”

When Weber was asked to handle replay reviews, he approached it in the same manner he did stock reports or scouting reports: He dived into research. He studied success rates of N.F.L. challenges and spoke with teams at both ends of the spectrum, as well as someone at the N.F.L. offices. In that first season, seemingly everyone in baseball had a different approach. The Oakland A’s used color-coded, numbered cards from 1 to 5 to signal to Manager Bob Melvin whether to appeal: A red 1 meant no appeal, a green 5 meant absolutely, and a yellow 3 was a coin flip.

(From the seventh inning on, the umpires can initiate a replay review on their own.)

“There are some philosophies that say challenge everything; if you lose, you lose it,” Weber said. “I go back to, it’s either right or wrong. I think our process works, but we continue to tweak it.”

Manager Joe Girardi added: “It’s not like you get as many as you want, so if you call for a replay, you better be right. Webby is usually right on. I don’t even run back and look, and I could run back and look in our clubhouse. I trust him with anything.”

That trust was reinforced last July 3 when Tampa Bay’s Evan Longoria hit a ball into the left-field corner in the 11th inning and slid into second base ahead of the tag by Jose Pirela. Umpire C. B. Bucknor signaled safe.

But Weber recognized on the slow-motion replay what was not recognizable to the naked eye: When Longoria popped up from his slide, he lost contact with the base for an instant while Pirela kept the tag on him.