With miscalled strikes, the same thing happens in reverse. A pitch in the strike zone thrown by a journeyman has a 19 percent chance of being called a ball. For the All-Star, the probability drops to 17 percent. That may not seem like a lot, but it means that an All-Star gets an automatic 9 percent advantage based not on his performance but on his reputation.

Although the effect is somewhat weaker, the same kind of bias appears when an All-Star batter is at the plate. The researchers calculate that a five-time All-Star has roughly a 5 percent or 6 percent advantage in getting a favorable ball-strike call.

The suspicion that the stars are treated differently by umpires is widespread among both players and fans, but until now it has been based on anecdote and rumor.

That there are numbers to support the belief came as a revelation to some players.

“I’m surprised at that,” the Mets’ Bobby Abreu said when told of the finding. Then he started to think. “Well, most All-Star pitchers are always around the plate,” he said, “so most of the time they’re going to get credit for a strike. Umpires know who’s on the mound.”

Al Clark, who umpired in the major leagues for more than 25 years before his retirement in 2001, said that on the contrary, umpires pay no attention to who’s who. “We don’t see who’s pitching; we don’t see who’s hitting,” he said. “We have to make a snap decision on what we see.”

And yet a few minutes later, Clark acknowledged that pitchers known for their control are in fact treated differently. “If a pitcher is throwing strikes, then it’s accepted that the zone is expanded,” he said. “If he’s not, he’s got to throw a defined strike.”

Mets reliever Vic Black is a pitcher who is likely to be a victim of the bias. “Really?” he said. “You mean the rich get richer? I never really thought about that. I just try to throw the ball over the plate. I know a lot of umpires miss a lot of my calls; sometimes they give me one.”