When the Gallup Organization studied 4,583 call center representatives from a major telecommunications company, researchers discovered that three representatives managed to alienate every customer on a single day — frustrating people so much that the company lost their business, possibly forever.

When those employees call in sick, their bosses no doubt look to the heavens and whisper, “Thank you.”

Co-workers and customers probably do, too. We’ve all been around toxic colleagues and know how companies frequently underestimate the harm that those people do. Unlike someone who is merely unproductive, the damage from these people can be more extensive than simply taking their salaries and throwing the money out the window.

“They don’t factor in the impact that the one person has on other people,” said Tom Rath, a global practice leader for Gallup. “The evidence is clear that one person can bring you down, day in and day out.”

Rath tries to emphasize how companies and workers can be upbeat in his new book, “How Full Is Your Bucket? Positive Strategies for Work and Life” (Gallup Press), which he wrote with his grandfather, former Gallup Chairman Donald Clifton. Just as they were finishing the book a year ago, Clifton died.

In that same telecommunications company survey, Rath added, seven of the service representatives were able to engage every customer they spoke with.

“Maybe you’ve been lucky enough to talk with a rep like this — one who listened to your problem, made sure you understood that you were heard, took care of your issues promptly, and left you feeling like he or she really cared about you as a person,” the authors write. “Did you want to tell others about this first-class service? And are you still a customer to this day?”

In a telephone interview, Rath explained how studies have shown that people and workgroups are more effective if they have more positive than negative interactions at work. “If you’re above 3-to-1, work teams are more productive than if they’re below that.”

Gallup has found that people who have a “best friend” at work are often more engaged and loyal, Rath said. A lot of that is because they have more positive interactions.

He is not suggesting, however, that companies ignore negative issues and pretend everyone and everything is wonderful. Rath said the studies also showed that such an emphasis on being positive can backfire if the ratio of positive to negative interactions is more than 13-to-1. Then again, not very many workplaces have to worry about even coming close to that ratio.

Although the book is more limited and not as insightful as some of Gallup’s earlier ones — particularly “First, Break All the Rules,” still the best book for bosses that I’ve come across — there are some useful anecdotes that managers in particular should pay attention to.

One points to a 1925 study conducted by Elizabeth Hurlock, who took fourth- and sixth-grade students and had them take math tests. While children in one group were praised in front of the class for their work, the members of a second group were criticized in front of their peers.

There were two other groups: One stayed in the class but was neither praised nor criticized, and the other, a “control” group, got moved to another room after the first test.

By the end of the fifth day, Hurlock found that those students who were praised showed a 71 percent improvement, compared with 19 percent for those who were criticized and 5 percent for those who were ignored. And all three groups did better than children in the control group, who heard neither praise nor criticism of anyone’s work.

But too many managers still act as if criticism is the important element to offer employees, particularly at performance review time, Rath said. Instead of looking at an employee’s strengths and figuring out how to make the most of them, the bosses dwell on flaws that in many cases could be compensated for by adjusting the job.

“It’s a lot better investment of time and money to focus on what you’ve done right,” Rath said.