The biggest advantage of a charter schools is that bad ones are easier to shut down than a public school. Detroit has some of the best and some of the worst. Cesar Chavez Academy is producing ivy league caliber students while the one that was run by former city council president Charles Pugh has all kinds of problems.
The Ben Carson high school appears to be public but students have to apply and keep a high grade level to get in or stay.
Charter schools are technically public schools