Looking at the data for the past five years, applying to more than 3 schools did NOT increase the acceptance rate. I made a spreadsheet a few months ago to analyze the data. As to how this is true, I'm not quite sure yet, but I do have some speculations. I am NOT saying that you only need to apply to 3; I just thought it was interesting and wanted to speculate why this may be true.
For the purposes of the analysis, I considered one offer from any school to be a success.
Those that only applied to one school were offered an acceptance at a rate of around 50%. Some of these may have been from SCCO's early application program (you apply exclusively to SCCO almost two years before you matriculate), and I think that's the only reason why anyone would just apply to one school. Then when we look at applicants that apply to two schools, the rate increases to just over 60%. This means that about 40% of applicants that applied to only two schools were rejected from both schools. At three schools, about 30% are rejected, and above three schools, the rate hovers around a 27% rejection rate, give or take 7 percentage points, no matter how many schools you apply to.
If you apply to EVERY school, like eight people have done in the past five years, you'll only get in about half the time. FOUR people have applied to every school and STILL got rejected from all of them.
It would make sense that applying to more schools will increase the chances of getting in, right? The data may suggest otherwise; as long as you apply to 3+ schools, your chances of getting in to at least one school doesn't increase. However, correlation =/= causation, so coming to this conclusion without further investigation is premature.
Another hypothesis is that those that apply to 5+ schools may have a weaker application compared to 3 school applicants, so while the rejection rate stays the same, the applications may be weaker overall. In other words, if we take the applications of those applying to 5+ schools and have them apply to only 3 of those schools, we should see an increase in rejection rate. This hypothesis makes more sense to me.
Bottom line: if you have a weaker application, you may need to apply to more schools to increase the likelihood that you apply to a school that accepts you.
As for schools knowing which other schools you've applied to, all I know is that they get to look at a master list to see where else you've been accepted. If they ask for it in the application (like PUCO does in their supplemental), then I think that's the only way they would know outside of you actually telling them where you've applied.
