Well the vast majority of mentally or emotionally disturbed persons will not attack unless provoked. I work in a small city that is the center for out patient Mental Health programs for half of 3 different counties (the city is smack dab on the border of the other 2). In terms of unprovoked violence they are the absolute LEAST of my concerns and I am the guy that gets called via 911 for the "altered mental status, advise EMS when the scene is secured" call.
They are NOT going to attack you in your home, unless they are a friend or a family member. If encountered on the streets they will rarely, if ever, attack if unprovoked.
http://www.apa.org/news/press/releases/2014/04/mental-illness-crime.aspx As a matter of fact, as a matter of statistics the mentally ill are more likely to be victims of violent crime than perpetrators.
Study Shows Mentally Ill More Likely to Be Victims, Not Perpetrators, of Violence | NC State University
The fear of those with mental illness is largely driven by mass shootings because most mass shooters etc end up having some form of mental illness but lets look at it from a numbers game. In 2015 there 372 mass shootings. Most of them were crime/gang related actually but for the sake of this exercise we will say they were all perpetrated by the mentally ill. (you can look them up here
Mass Shootings). They killed over 400 people and wounded over 1000 people, in a population of over 320 million.