On Monday, Eric Adams announced a plan to combat gun violence in New York City, committing $485 million to a slate of initiatives largely aimed at prevention — more job training and expanded mental health care among others. It would focus on six precincts in the Bronx and Brooklyn, where nearly a quarter of the city’s shootings last year took place, a seemingly wise, data-driven use of resources.

At the news conference, a reporter asked the mayor how he could square his refrain that New York remained “the safest big city in America” with the reality that people didn’t necessarily feel that way. What would this program do, she wondered, to reduce “the random acts of violence that make people believe that they are not safe”?

Since the mayor established a gun violence task force last year, there has been less cause for anxiety, at least by statistical account: Shootings in the city are down 26 percent so far this year compared with last, with murder declining by 11 percent. Burglaries and robberies are also less frequent.

But the question of how bad crime actually is versus how bad it is perceived to be is complicated by the occurrence of rare but terrifying incidents, like the death of Michelle Go, who, in January last year, was pushed onto the subway tracks by a homeless man in Times Square at 9:30 on a Saturday morning.

These sensational moments, compounded by history and exposure — the notion that if some acts of violence are random they can randomly happen to you — turn the walls of reason porous and flood us with emotion. They are attenuated, to some extent, by Mr. Adams’s own rhetoric, which has often lacked the reassuring effects of clarity.

In May of last year, for example, the mayor said that he had “never witnessed crime at this level,” even though he was very much around, working as a transit cop, in the early 1990s, when the murder rate was more than four times as high. Then, last month — and not for the first time — he blamed the fear of crime on the press.

The effect of blown-out coverage of the most horrific events “plays on your psyche,” he said in a television interview, not inaccurately. But the judgment came after a Siena College poll showed that 39 percent of those surveyed in the city said that they had “never been this worried” about personal safety as they were “today,” essentially echoing the mayor’s own hyperbole.

Although the poll was based on a very small sample size — it was conducted statewide, in June, with only 345 city residents questioned — it delivered the image of a place operating at high-frequency alarm. “The level of concern across the whole sample in New York City was dramatically high,” Don Levy, director of the Siena Research Institute, said.

Seventy percent of those living in New York City indicated that they were concerned about becoming the victim of a crime, and 17 percent said they had bought a firearm for self-defense during the past 12 months, nearly twice the rate of those surveyed upstate, even though both violent and property crimes are proportionally higher in places like Buffalo and Syracuse than they are in New York City.

Even when fear seems entirely justified, its expression can be hard to understand, inspired by so many variables. This is one of several resonant lessons from Leon Neyfakh’s excellent new podcast, “Fiasco: Vigilante,” which revisits the 1984 case of Bernhard Goetz, who shot and wounded four Black teenagers (leaving one paralyzed) on the subway after one of them demanded money from him and he concluded that he was about to be mugged.

An early episode of the series cites the research of the criminologist Dennis Kenney, who, during those years when violence was so commonplace in the city, surveyed riders across four sectors of the transportation system. What he found was that riders were not afraid in the moment. In fact, a person’s chance of being victimized during any single ride was almost nonexistent — .001 percent.

But many assumed they would be mugged or assaulted at some point, and this feeling of “inevitable victimization’’ stemmed from the sense that no one was in charge, that the city was in free fall, which in turn created a kind of incubation space for vigilantism. Three percent of those interviewed in 1984 said that they were carrying a gun, a striking statistic given the strict gun laws of the time.

A poll conducted by The New York Times in January 1985, a few weeks after the Goetz shootings, found that half the city’s residents believed that crime was the worst thing about living in New York. Nearly as many felt that it was sufficiently terrible to justify taking matters into their own hands. A majority expressed support for what Mr. Goetz had done. He was ultimately acquitted on charges of assault and attempted murder and sentenced to six months in prison for carrying an illegal concealed pistol.

In the days after the shootings, while Mr. Goetz was at large and the tabloids pumped out day after day of panic-feeding content, the “subway vigilante” emerged as a folk hero to New Yorkers. After he surrendered to police in New Hampshire and was returned to the city, Joan Rivers sent him a message of love and support at Rikers. Some offered to cover his $500,000 bail (he refused); others suggested that he run for mayor.

In response, Benjamin Ward, the city’s first Black police commissioner, argued, however unconvincingly, that “the perception of crime” was “greater than the reality,” and that major crimes had been in decline since 1981. But the city was still reeling from the murder of a 23-year-old Harvard graduate and actress named Caroline Isenberg, who was stabbed to death on the roof of her West End Avenue building three weeks before the Goetz shootings and after she had rejected the sexual advances of her assailant.

Three months ago, Daniel Penny’s chokehold killing of Jordan Neely, a 30-year-old homeless Black man he encountered on the subway, drew comparisons with the actions taken by Goetz, so many years ago. The aftermath, however, was very different.

Many were outraged by the ex-Marine’s response. But in the ’80s, the columnist Jimmy Breslin seemed like an outlier in calling out Mr. Goetz’s misplaced reaction to an imagined threat.

Near the end of “Fiasco: Vigilante,” we hear from Raven Cabey, sister of Darrell Cabey, the teenager whose spine was severed by one of Mr. Goetz’s bullets. She explains that her brother long ago left the city and doesn’t like the idea of even coming to visit. For all that Mr. Goetz seemed to hate about New York, viewing it a cesspool of crime and disorder, he nonetheless remained in Manhattan for decades.