When Rep. Gabrielle Giffords, D-Ariz., announced that the Obama administration would send as many as 1,200 additional National Guard troops to bolster security at the Mexican border, she held up a photograph of Robert Krentz, a mild-mannered rancher who was shot to death this year on his vast property. The authorities suspected that the culprit was linked to smuggling.

"Robert Krentz really is the face behind the violence at the U.S.-Mexico border," Giffords said.

It is a connection that those who support stronger enforcement of immigration laws and tighter borders often make: Rising crime at the border necessitates tougher enforcement.

But the rate of violent crime at the border, and indeed across Arizona, has been declining, according to the Federal Bureau of Investigation, as has illegal immigration, according to the Border Patrol. While thousands have been killed in Mexico’s drug wars, raising anxiety that the violence will spread to the United States, FBI statistics show that Arizona is relatively safe.

That Krentz’s death nevertheless churned the emotionally charged immigration debate points to a fundamental truth: Perception often trumps reality, sometimes affecting laws and society in the process.

Judith Gans, who studies immigration at the Udall Center for Studies in Public Policy at the University of Arizona, said that what social psychologists call self-serving perception bias seemed to be at play. Both sides in the immigration debate accept information that confirms their biases, she said, and discard, ignore or rationalize information that does not. There is no better example than the role of crime in Arizona’s tumultuous immigration debate.

"If an illegal immigrant commits a crime, this confirms our view that illegal immigrants are criminals," Gans said. "If an illegal immigrant doesn’t commit a crime, either they just didn’t get caught or it’s a fluke of the situation."

Gans noted that sponsors of Arizona’s controversial immigration enforcement law have made careers of promising to rid the state of illegal immigrants through tough legislation.

"Their repeated characterization of illegal immigrants as criminals — easy to do since they broke immigration laws — makes it easy for people to ignore statistics," she said.

Moreover, crime statistics, however rosy, are abstract. It takes only one well-publicized crime, like Krentz’s shooting, to drive up fear.

It is also an election year, and crime and illegal immigration — and especially forging a link between the two — remain a potent boost for any campaign. Gov. Jan Brewer’s popularity, once in question over promoting a sales tax increase, surged after signing the immigration bill, which is known as SB 1070 but officially called the Support Our Law Enforcement and Safe Neighborhoods Act.

No matter that manpower and technology are at unprecedented levels at the border, it may never be secure enough in Arizona’s hothouse political climate when congressional seats, the governor’s office and other positions are at stake in the Aug. 24 primaries.

It took the Obama administration a few weeks to bow to that political reality and go from trumpeting the border as more secure than it had ever been to ordering National Guard troops to take up position there — most of them in Arizona, Obama assured Brewer in a private meeting — because it was not secure enough.

Crime figures, in fact, present a more mixed picture, with the likes of Russell Pearce, the Republican state senator behind the immigration enforcement law, playing up the darkest side while immigrant advocacy groups like Coalicion de Derechos Humanos (Human Rights Coalition), based in Tucson, circulate news reports and studies showing that crime is not as bad as it may seem.

For instance, statistics show that even as Arizona’s population swelled, buoyed in part by illegal immigrants funneling across the border, violent crime rates declined, to 447 incidents per 100,000 residents in 2008, the most recent year for which comprehensive data is available from the FBI. In 2000, the rate was 532 incidents per 100,000.

Nationally, the crime rate declined to 455 incidents per 100,000 people, from 507 in 2000.

But the rate for property crime, the kind that people may experience most often, increased in the state, to 4,082 per 100,000 residents in 2008 from 3,682 in 2000. Preliminary data for 2009 suggests that this rate may be falling in the state’s biggest cities.

What is harder to pin down is how much of the crime was committed by illegal immigrants.

Phoenix’s police chief, Jack Harris, who opposes the new law, said that about 13 percent of his department’s arrests are illegal immigrants, a number close to the estimated percentage of illegal immigrants in the local population. But the Maricopa County Sheriff’s Office, which runs the jail for Phoenix and surrounding cities and is headed by Joe Arpaio, a fervent supporter of the law, has said that 19 percent of its inmates are illegal immigrants.

Scott Decker, a criminologist at Arizona State University, said a battery of studies has suggested that illegal immigrants commit fewer crimes, in part because they tend to come from interior cities and villages in their home country with low crime rates and generally try to keep out of trouble to not risk being sent home.

But he understood why people’s perceptions of crime might lag behind what the statistics show. "Hard as it is to change the crime rate, it may be more difficult to change public perceptions about the crime rate, particularly when those perceptions are linked to public events," Decker said.

He added, "There is nothing more powerful than a story about a gruesome murder or assault that leads in the local news and drives public opinion that it is not safe anywhere."

Kris Kobach, a University of Missouri law professor who helped write the Arizona immigration law, pointed to crimes like a wave of kidnappings related to the drug and human smuggling business in Phoenix, something Brewer herself noted when she signed the law.

Although the reports have dipped in the past couple of years, the police responded to 315 such cases last year.

"That’s scary to people, and people react to that all over the state," Kobach said. "They are concerned. ‘That might happen in my part of the city eventually.’ "

Terry Goddard, the state attorney general, who does not support the immigration law, said the drop in violent crime rates might not reflect the continued violence, often unreported, that is associated with smuggling organizations.

Goddard said he doubted that the immigration law would put a dent in the smuggling-related crime that grabs attention in the state. For that reason, Goddard, who is running to be the Democratic nominee for governor in the primary, said he backed the deployment of National Guard troops and supports increasing manpower and spending on police and prosecutor anti-smuggling units.

Brian L. Livingston, executive director of the Arizona Police Association, said he would prefer more attention on the border, too. But until then, he said, laws like Arizona’s are necessary.

"We know the majority of people crossing across are not criminal, but unfortunately some criminal elements are embedded with them," he said, adding, "Governor Brewer gets that."

As Brewer put it just after signing the bill: "We cannot sacrifice our safety to the murderous greed of drug cartels. We cannot stand idly by as drop houses, kidnappings and violence compromise our quality of life."

© 2010 The New York Times Company