Contrary to popular belief Schizophrenia technically isn't one disorder, its an umbrella of disorders everything from Classic Schizophrenia to Schizoaffective Disorder which i have. Schizophrenia can be split into two words Schizo = Split, and Phrenia = Mind, but this doesn't mean split personality, or a literal splitting of the mind, it means a fragmentation between reality and the mind, to Schizophrenics the line between what happens internally, in their mind and what happens in the outside world, reality, is very very blurry and they can't tell the difference.
--What is Schizophrenia and Schizoaffective Disorder, what are the differences?--
Classic Schizophrenia is a mental disorder in where the person struggles to hold on to reality, often Schizophrenics suffer from Delusions and Hallucinations where they hear things that aren't there or believe things that aren't true, the most common form of Hallucinations is hearing voices that to the person sound real but are in fact a figment of the person's mind, and Delusions like believing the person is being spied on by someone or the government. People with Schizophrenia are perfectly capable of living perfectly healthy and normal lives so long as they stay on the medications and therapies that they need to keep the symptoms down, people with Schizophrenia ARE NOT more likely to be serial killers, actually they are more likely to harm themselves than other people. Since once they become aware that the voices are internal, the anger they have becomes internalised. Classic Schizophrenia on its own, doesn't usually lead to any change in their mood, their mood is changed by the experience of Positive Symptoms (Things that are added to the system rather than part of it such as Hallucinations) but mood changes aren't typically part of the Schizophrenia diagnosis.
Schizoaffective Disorder is a mix of Schizophrenic symptoms and Mood disorder symptoms. People like me with Schizoaffective Disorder experience very similar if not identical psychotic symptoms like Hallucinations, Delusions and disorganised thoughts/speech with mood disorder symptoms like Mania, Depression and Mixed (Much like Bipolar disorder). Schizoaffective people again like Schizophrenics are fully capable of living perfectly healthy lives, with help from Mood stabilising medication and therapy (most commonly CBT).
--Why do people have this idea that Schizophrenic people are Serial killers?--
Schizophrenia and its sibling disorders are some of the most misunderstood mental illnesses, I've met people who when they learn i'm Schizoaffective they change their attitude and act like they are walking on eggshells around me. I'm not going to hit or kill anyone, the voices i get aren't telling me to do anything that i wouldn't do on my own.
Well the key source of this false belief is partially from old serial killer cases where the Serial killer would often say "The voices told me to do it" and thus a false equivalency started that if serial killers hear voices, and Schizophrenics do as well, then surely Schizophrenics are Serial killers. However there is no evidence that Schizophrenics have a higher likelihood of being arrested or jailed for violent crime, Yes Schizophrenics do have violent outbursts, but a lot of the time its internalised to self-harming and suicide attempts, not mass killing spree's.
The other source is how movies, most commonly Horror Movies have shown all of their Serial killers to have Schizophrenic symptoms, but most people like with most things, when they see something like that being shown to them all the time, and they never see anything different from movies. Then they are more likely to believe that is true, but this point almost fades into the first point.
--How should you be around Schizophrenics and Schizoaffective people?--
Yourself is the easy answer, don't be on eggshells around us. That's making you seem more suspicious not less, if you want to know what to avoid doing or saying around a Schizophrenic, then ask us. Same thing with people who suffer from PTSD, if there are trigger points that the person is aware of, just ask them. And most importantly, SUPPORT THEM, Don't turn your back or run away from Schizophrenics, we're people just like you. If you were scared and felt alone, the last thing you would want someone to do is turn their back and walk away from you when you most needed it.