When it comes to mass shootings, President Barack Obama and House Speaker Paul Ryan are in rare accord on a leading culprit.

Both point fingers at mental illness. And in poll after poll, most Americans agree.

But criminologists and forensic psychiatrists say there is a flaw in that view: It doesn't reflect reality.

While acknowledging that some of the country's worst mass shooters had varying degrees of mental illness — the Colorado movie theater shooter, James Holmes, with his orange-dyed hair; the Virginia Tech shooter, Seung-Hui Cho, whom a judge ordered to get treatment — experts say the vast majority of these killers did not have any classic form of serious mental illness, such as schizophrenia or psychosis.

Instead they were more often ruthless sociopaths whose behavior, while unfathomable, can't typically be treated as mental illness.

The oversimplification, experts say, is perpetuated by the gun industry and a society that assumes that the mentally ill are the only ones capable of vicious killing sprees. Now, with the White House and Congress prioritizing an overhaul of the mental health system to try to curtail mass shootings and gun violence, critics say the country is chasing an expensive and potentially counterproductive cure on the basis of the wrong diagnosis.

“It would be ridiculous to hope that doing something about the mental health system will stop these mass murders,” said Michael Stone, a forensic psychiatrist at the Columbia College of Physicians and Surgeons and author of “The Anatomy of Evil,” which examines the personalities of brutal killers. “It's really folly.”

Stone maintains a database of more than 300 killers, most of them shooters of four or more people. He breaks mental illness into two categories. Those with schizophrenia, delusions and other psychoses that separate them from reality and who are suffering from serious mental illness and could be helped with medical treatment. And those with personality, anti-social or sociopathic disorders who may exhibit paranoia, callousness or a severe lack of empathy but know exactly what they are doing.

In a paper published last year, Stone found that just about 2 in every 10 mass killers had serious mental illness. The rest had personality or anti-social disorders or were disgruntled, jilted, humiliated or full of intense rage. They were unlikely to be identified or helped by the mental health system, reformed or not.

These traits, by Stone's analysis and definition, describe Eric Harris, the ringleader of the Columbine High School shootings in 1999; Michael McDermott, a software technician who killed seven co-workers in 2000; Nidal Hasan, an Army major who killed 13 people at Fort Hood, Texas, in 2009; Dylann Roof, who is charged with killing nine worshippers in a church in Charleston, S.C., last year; and many more.

“The whole notion of mental illness and mass shootings is so poorly understood,” said Liza Gold, a forensic psychiatrist at Georgetown University and editor of a recent collection of scholarly papers on gun violence and mental illness. “To address the reality of it, it's like dealing with people in a parallel dimension.”

Around the country, at the federal and state levels, lawmakers have proposed or passed legislation linking mental illness to gun violence, saying the measures were needed to stop mass shootings. Some states, including New York, now require mental health workers to report anyone they think is dangerous to a database used for firearms background checks. After the Virginia Tech shootings in 2007, Virginia passed measures to lower the criteria for commitment to a mental health facility.

Almost every recent high-profile mass shooting in recent years has prompted plans and promises to reform the mental health system.

Jonathan Metzl, a Vanderbilt University professor who studies the history of mental illness, has written that “insanity becomes the only politically sane place to discuss gun control.”

Earlier this year, in a tearful announcement of measures to stem the American phenomenon of mass shootings and gun violence, Obama said, “We're going to do more to help those suffering from mental illness get the help that they need.”

He proposed spending $500 million to expand mental health treatment. His frequent legislative sparring partner on Capitol Hill — the House speaker — is insisting that something be done to protect Americans from randomly being shot.

“We have seen consistently that an underlying cause of these attacks has been mental illness, and we should look at ways to address this problem,” Ryan, R-Wis., said earlier this year.

The question is how.

In 2013, in response to the shootings at Sandy Hook Elementary School in Newtown, Conn., that killed 20 first-graders, Rep. Tim Murphy, R-Pa., a child psychologist, proposed a far-reaching mental health reform package to expand inpatient psychiatric care and relax privacy rules so family members of the mentally ill are able to access their health records. States would have lost federal grant money if they didn't pass laws forcing people to get outpatient treatment. The bill, which had Democratic co-sponsors, stalled amid concerns about patients' privacy and the involuntary treatment provision.

Murphy revised and reintroduced the bill last year, backing off the involuntary treatment requirement. Sen. Chris Murphy, D-Conn., also has proposed a reform package, although he has not linked it to mass shootings, saying he was “uncomfortable having mental health framed as a response to gun violence because it risks drawing an inherent connection between mental illness and violence, which doesn't exist.”

The senator's bill is competing with legislation sponsored by Sen. John Cornyn, R-Texas, that would require courts, not mental health professionals, to determine whether someone should be prohibited from buying guns. Democrats say that would make it tougher to seize firearms, but the National Rifle Association supports Cornyn's measure.

There may not be much chance of any meaningful reform passing in an election year. Still, both parties seem determined to get something done.

“The reality is, so many of these mass shootings could have been prevented,” Rep. Murphy said in an interview. “The issue is identifying these people sooner and getting them the help they need.”

But psychiatrists and criminologists who specialize in mass killings say these cumbersome and expensive efforts will have little effect in stopping mass shootings. They fear that the country will be given a false sense of security and that when the shootings persist, the mental health system will be blamed again.

Critics are especially concerned about increased stigmatization of the mentally ill, fearing that they will avoid treatment so their medical records aren't entered into databases, some of which have derogatory category titles such as “the mentally defective file.”

Underlying the disconnect between the legislative ideas and the scientific reality are, experts say, fundamental misconceptions about the connection between serious mental illness and violence.

Studies show that the mentally ill do present a higher risk for violence than others, but overall they account for just 3 to 5 percent of violence in the country — and only 1 percent of gun violence against strangers. They are far more likely to be victims of crime.

There are many groups perpetuating the myth of the mentally ill mass shooter, experts say.

One is the news media, which looks for and raises the mentally ill storyline after major incidents, sometimes without confirmation but with profound effects. Readers of news articles linking mental illness to a mass shooting “reported significantly higher perceived dangerousness of, and desired social distance from, people with serious mental illness in general,” according to a paper by researchers at Duke and Johns Hopkins universities.

Another is the NRA, whose officials, in fighting off tighter gun control policies, have called mass shooters “so deranged, so evil, so possessed by voices and driven by demons that no sane person can even possibly comprehend them.”

And most Americans agree, with 63 percent blaming mass shootings on the failures of the mental health system to identify sick people before they act, according to a Washington Post-ABC News survey last year.

“I think it's the human inclination to explain behavior that is frightening and tragic as the result of mental illness, because it's very hard to understand that individuals do not have to be mentally ill to do something frightening and tragic,” said J. Reid Meloy, a forensic psychiatry professor at the University of California at San Diego who studies mass killings and consults with the FBI.