If you might be interested in a purely undergraduate-focused environment, these smaller colleges appear in a Princeton Review category, Great Schools for Mathematics Majors:
St. Olaf
Holy Cross
Macalester
Reed
Grinnell
Harvey Mudd
Hamilton
Bowdoin
Haverford
Carleton
@CallieMom It seems both Mit and Harvard have interest marked “not considered” while Princeton has it marked considered. The common data set is very useful. I was planning a visit there already actually…
You can check whether “level of applicant’s interest” is considered in section C7 of the college’s common data set, or in the admissions tab in its entry on http://www.collegedata.com .
Colleges usually consider that if they are not interested in being someone’s low-choice “safety”, so it is more of a concern if it looks like you are “overqualified” for the college.
http://talk.collegeconfidential.com/college-admissions/1626043-ways-to-show-a-high-level-of-applicants-interest-p1.html is a discussion about showing interest.
The definition of “safeties” varies. Some people here like to think of them as true sure things, thus the automatic admission criteria, understanding their FA in advance, etc. That’s perfectly fine, but many others use safety much more broadly to mean simply extremely likely. If you are going to make safety 100% guaranteed, you almost need a fourth category for the extremely likely that are different than the targets. None of my S’s friends nor his school’s counselor interpreted “safety” as guaranteed. What they did do is make sure to apply to more than one safety just in case they didn’t get accepted to one. I didn’t hear about a single kid at my S’s school for whom this didn’t work out and they didn’t get in to any of their safeties. Of course you do hear about people nationally who don’t get into any schools but when you look at their safeties they usually were targets to reaches by any objective measure.
Anyway, all that said, from your original list I would call Case a safety and USC a target based on your stats and reviewing the typical GPA and score data. Case was one of my S’s safeties (added at the last minute when they waived the application fee and required no supplements) and he was accepted and offered a very generous merit scholarship. His GPA/scores were not better than yours. Some people will say no school that accepts only 30-some percent can be a safety. All of my S’s were and he got into all of them and was offered merit aid at all that offer it. He applied to 3 just in case.
The Common Data Set is awesome. Many interesting things in it. I will caution though that schools are elective in filling it out and some don’t due so comprehensively. So their data is not always complete, including how many graduates per major. We encountered this pretty frequently where if you added up all the degree stats they were a fraction of the graduation rate.
@ucbalumnus For a friend: in your opinion, or as far as you know. How much would a score increase of 1 from
33–>34 improve an individuals chances at a top school on average( talking harvard/mit level, schools with ranges 33-35)? Can something like that really be measured? Is it insignificant?
Why are Your colleges all reach? Where are your safeties?
@NASA2014 The whole post is about help with a college list. I ask in the post if certain schools are safeties. I wasn’t even sure what the true definition of a safety was. Also, safeties are just less interesting to look for.
Your safety(ies) are probably the most important school(s) on your list. A safety should be a place you can definitely get in, can afford, and would be HAPPY to attend. If it’s not all three then it’s not a safety.
I have some familiarity with UIUC, and I’d say, if the qualifications listed in the OP are accurate/representative, this student is very likely to get in (getting in to the College of Engineering would be less clear, but Mathematics is in LAS at UIUC). I also agree that this student is very likely to get into Case Western. (Yes, I know, that and a few bucks will get you a cup of coffee …)
That said, it’s still a good idea to consider other “safe” schools. For STEM-oriented students in Illinois, some possibilities are Purdue, Wisconsin, Minnesota, Iowa State, MUST, and Rose-Hulman.
Regarding safeties, the general advice here is that they’re the most important schools on the list, or put another way, build your list from the bottom up. I agree that safeties are very important, but my take is a little different. I’d say that matches are the most important – in that they’re probably the largest group on the list, and that the most time will be spent considering them. But there are a couple of corollaries that are very important here – that you go into the process being able to assess things fairly objectively, reasonably, and realistically, and that you plan for various contingencies, even some fairly unlikely ones. And that will still necessitate including a few “safe” schools.
So, OP, I think you need to add a few matches and maybe a couple of safeties.
Another thing regarding safeties, I vaguely recall that when I first started visiting this site a few years ago, people would list four criteria for determining a safety, not three (as in comment #27). I can’t remember what the fourth one was, maybe something like the school had to have an acceptable major available? (Or maybe my memory is just going 
).
@csdad2 Thanks for the advice
My advice would be somewhat contrary to @csdad2’s. If you do find a true safety then you don’t need matches. You can apply to reaches to your heart’s content knowing that you’d be happy at your safety college.
^not if you’d prefer the match
@“Erin’s Dad” By that logic, why would you even need reaches if you have a perfect safety?
In our experience, preference was on a continuum, so even if our S tried to find a couple safeties he was happy with that didn’t mean there were not schools that were more selective that he wanted to attend more. Some of those we called reaches, some of them matches, just depending on how far beyond of his stats (relative to his school’s past performance) he was. Settling on a couple (just in case) safeties was very important, but he put the most focus on matches because those were the schools he still had a decent shot at getting into that he preferred over his safeties.
Agree with @csdad2 and @wayneandgarth. Matches, if chosen well, may offer excellent chances for admission (and therefore collectively can serve as a near safety), while high reaches may, in aspects, be somewhat clichéd “usual-suspects, someone-else’s-type-of-school” places – and often may not be particularly intrinsically desirable in any event compared to carefully chosen match schools.
No one said that a safety has to be perfect. But the safety needs to be liked enough so that, if the student’s only affordable admission is to the safety, s/he will not be feeling a huge let-down, disappointment, or sense of failure.
If the student’s application list has no actual safety, the default safety is to start at a community college or some such.
@ucbalumnus We’re on the same page about safeties. I certainly get their value and said to. I was responding to the comment above that matches are not important if you have a great safeties. My point was that just because a person has a great safety doesn’t mean they wouldn’t potentially like their matches even more, same as with the reaches.
I think it is always nice to have choices in the spring, hence the desire for matches. Also, I think accepted student visits sometimes reveal more about the college – it is possible that a safety might be less desirable after that experience. And some kids change their minds or preferences between January and May 1 – again, more than one choice is good in the spring.
@intparent @citivas @merc81 @ucbalumnus @“Erin’s Dad” Thanks for all the comments. In the process of searching for schools, I knew I wanted to go reach heavy; after just a few visits, I knew that the people/environment/rigor were what I wanted in a college. But I haven’t really found an effective way of determining my chances to any given school. Of course there are Score ranges and things like that, but a school like Mudd for example has a range very similar to MIT (again for example) but has a very different acceptance rate. I have come across certain chance calculator websites, but they all tend to operate under a very small data sample ( 100 students for example, and the sample is of course not representative of the average applicant, it is comprised of individuals who have submitted their scores and results to the website) The best thing I have come across is PrepScholars Admission calculator tool. It is part of their “college database”. Here is a link to their page on UChicago…
http://www.prepscholar.com/sat/s/colleges/University-of-Chicago-admission-requirements
They claim that their calculations are based on data collected from thousands of students (again, probably not representative of the average applicant)
I wanted to ask if their is a true way to calculate reasonably accurate admission chances. (within say 5% error)
Is the site I linked accurate to the best of your knowledge? Is there a website or service that can give better odd calculators? This is important to me to judge what my safeties and targets (and reaches) really are.
Mudd’s acceptance rate isn’t that different from MIT’s. 8% MIT vs 13.6% Mudd last year, I think. And Mudd was 9% for male applicants.
You can’t know your chances. That is why people apply to more than one school. None of those tools giving “chances” have a solid set of data. If your school has Naviance, that is a decent source. But discuss it with your GC, they know things like if accepted students with lower stats were hooked.
I used chancing calculators extensively for D16 - didn’t help. There was one school that all the calculators said she was >80% likely to get in. Her ACT score was several points higher than their 75th percentile. She was denied. I believe part of it was that she didn’t show “interest”. Honestly - paying $200 for an application and score sending and writing essays isn’t enough interest??? If we had to do it over again - I would make sure she showed lots of interest to every school, whether or not the common data set says the school values it.