Here's a link to the base rate fallacy article on Wikipedia: https://en.wikipedia.org/wiki/Base_rate_fallacy.
A thing to avoid in your situation is focusing excessively on the specifics that lead you to conclude that the local public school will be as good as the private school you are attending. Generally speaking, public schools are lower quality than private schools. But getting a little more narrow might be worthwhile: How are the public schools in your general area compared to public schools in the country, using objective statistics on things like SAT rates? Now, can you compare that to your private school or your brand of private school (Catholic, secular, whatever)? Think about other metrics that matter for you: percent getting into Ivy League colleges, or number of assaults on campus per 100 students, or whatever.
Compare those rates before mentally inserting yourself into the situation. Once you mentally place yourself there, a lot of what you know about the statistics of the places can slip away. That is what the base rate fallacy teaches. It helps you focus on the idea that the median experience at each school is likely to be your experience, which helps defeat the "grass is greener on the other side of the fence.".
UPDATE 3/16/16: I decided to go to public school, because I was tired of all the little annoying stuff at my current school--especially the entitled kids and the entitled attitude in general. Everybody acts like they deserve something. It's very irritating.
The other reason I came to that decision was "exploration value". By moving to a new situation I learn whether I really am better off in the kind of environment offered by the public high school; even if it ends up being worse for me, at least I know what to avoid. If it's good, I know it's good; and if it makes no difference, I know that, too.
EDIT: Just to clarify, I attended public middle school with the same group of kids, so I won't have to worry too much about getting to know new people. I am still in occasional contact with my old friends. I talked to one of them for several hours just yesterday.
(I'm new here, though I've lurked, so if I break any rules or otherwise do something detrimental, please let me know and I will try to correct my mistake)
I currently go to a rather nice independent high school. I'm on significant financial aid, so I can afford it. The academics there are outstanding. However, being a boarding school (I go as a day student) it requests a lot of our free time. We are required to participate in adult-sanctioned activities at least six or seven hours a week, in addition to normal classes. This means that 1. I get home from school around 6pm and 2. (more importantly) it's very hard to socialize when you don't board at the school, and there's really very little besides drama or sports (neither of which I like very much) that people do after school and actually enjoy/make friends in.
I'm strongly considering transferring to my public school, which is unusually good for a public school, as a junior next year. The academics are not as great (classes are less discussion-based and there are not as many APs offered) but there is a strong amount of participation in stuff I might actually enjoy after school (math team, etc.) because we're not required to do anything after school. I've noticed that when people tell me I have to do things, I enjoy them much, much less. Also, I won't have a commute, which would be nice.
Everything else about the schools is more or less comparable.
I'm sure when I think about this decision I am biased in some way. I'm probably succumbing to the sunk cost fallacy (sometimes I think if I'd been at the public school the last two years, I'd rather be at the private school) or something like that. If not, I'm facing the problem of Buridan's Ass.
My question to you: should I transfer or not? I have thought very hard and consulted several intelligent people, and have not been able to come to any sort of conclusion.