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 also 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 have 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,” Mr. 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 Ms. 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.