Searching for schools that are reasonable reaches based on SAT

College Search & Selection
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It will surely depend on how the student is doing in high school (4.0 GPA versus 2.7 GPA would mean different suggestions), and the parent financial constraints (dropping out of college because the student and parents ran out of money is a common reason for dropping out of college).

Actually, the role of SAT scores is quite variable across colleges. Some colleges treat it as much more or less important than other colleges. Assuming that all colleges treat it as the most important factor in admissions is a mistake, and sometimes leads to choosing “safeties” that are actually reaches, or reaches that are realistically out of reach.

62

From a mathematical point of view, what you are looking for is given by Bayes’ Theorem. It says that the conditional probability of event A given B (written as Pr(A | B)) equals Pr (B | A) times Pr (A) divided by Pr(B). You can see the formula displayed more clearly if you look at any online definition of Bayes’ Theorem.

In the case of college admissions, you are seeking the probability of acceptance to a given college (event A), given a particular SAT score (event B), among all applicants to that college. Therefore, you need to know the probability Pr(B | A) of getting that SAT score given acceptance, which is given by the percentiles you provided in your original post. You also need to know the probability of acceptance Pr(A), and the probability (essentially the fraction) of having that SAT score Pr(B) among the applicants to the college. Note that Pr(B) may take some effort to determine, since it is not often reported by colleges.

Let’s consider an example. Consider a student who has an SAT critical reading score of 600. Event A is therefore defined as having a score of 600 or below. (In probability, we generally consider events to be intervals (e.g., 600 or below), and not points (e.g., 600).) Consider the probability of getting into Stanford, based just on that information. I’m picking Stanford as the college, because it provides the necessary statistical quantities we desire on their Web site and printed materials. The information indicates that 20% of applicants had a CR score of 600 or below, and 5% of their admitted class had that score. Therefore, Bayes’ Theorem says to multiply that 5% times their overall admit rate of 4.8%, divided by 20%. The result is 1.2%. In fact, Stanford also directly provides the admit rate for that group of people as 1%, which of course matches the result of Bayes’ Theorem.

Hope this helps. In practice, you want the calculate the probability of acceptance, given all of the factors for a given applicant (GPA, SAT scores, essays, recommendations, and so on). Unfortunately, that calculation is complicated by the fact that the necessary marginal and conditional probabilities are not conditionally independent and not readily obtainable.

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Generic advice:
Aim for a balanced reach-match-safety list of affordable colleges you’d be happy to attend.
Then, when the results come in, pick the best offer.

For some students, “best” will indeed mean the most selective.
For others, it may mean the closest, the cheapest, or the only one with a specific program.

Over many thousands of these decisions, students probably do tend to choose the most selective school to admit them. If they truly preferred someplace less selective, there’d be little point in submitting a much less likely application. Nevertheless, many other factors besides admission selectivity affect the application preferences of well-informed students.

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If anything, the mid 50 score range is still far more informative that this. At least it is contributed by the main population of admitted/enrolled freshmen. Below is a link to a PDF with compiled mid 50 range from last year with the new SAT score scale and ACT.
http://downloads.■■■■■■■■■■■■■■■/Compass-360-College-Profiles-New-SAT-and-ACT.pdf

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@billcsho, if an applicant is in the middle of the mid-50 and doesn’t have any hooks, then as they go up the score profiles the likelihood of acceptance will diminish dramatically - this is why so many cry out that they didn’t get into any of their matches when the schools were really reaches, but none of their guidance group was honest with them about this important distinction.

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@Chembiodad I am not saying the mid 50 table tells everything. I said if anything we can get from the SAT score, the mid 50 information is far more informative than what OP posted. I believe the admission rate for each score tier is a better index. Go back and read my previous comment. I emphasize the importance of admission rate which OP does not believe.

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@pantha33m, agree. That’s why a single ED reach is fine if the student profiles in the top-25% at every other school they apply and they don’t have any hooks.

As we have all learned, an unhooked applicant that profiles in the top-25% at a school that accepts less than 20% isn’t very good odds. My DD’s had plenty of friends that overreached and are attending schools they never imagined they would be at - I think this is a symptom of braggadocio by parents as much as anything else.

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@billcsho, got it. Agree that the probability of acceptance decreases exponentially as you move up the selectivity index, to the point that when you get to the top all unhooked applicants scores are bunched together in the 34-36 ACT range and then both institutional and individual holistic factors come into play.

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I think that some mathematics might help to clarify things. (Forgive me if a duplicate post from me about this topic appears; somehow in the course of editing, my earlier post seems to have disappeared.)

You might recall from your probability class that a conditional probability Pr(A | B) is the probability of event A occurring, given that event B has occurred. The original post provided the percentiles of the SAT scores of the accepted applicants. Those are the conditional probabilities of achieving a particular SAT score, given that one has been accepted to a particular college. This is quite different from what we really want, which is the probability of acceptance, given a particular SAT score.

To show that these are different concepts, consider the following fallacy: 50% of the people accepted to college X are female. Therefore, if I am female, I have a 50% chance of being accepted to that college. Clearly, this form of reasoning is not valid.

However, the two conditional probabilities are related by Bayes’ Theorem. Bayes Theorem says that Pr(A | B) = Pr(B | A) * Pr (A) / Pr(B). In this context, let A be acceptance by a particular college, and let B be obtaining a particular SAT score. Then, aside from the percentile information of the accepted applicants, which is Pr(B | A), we need two additional pieces of information: Pr(A), which is the overall probability of being accepted to the college, and Pr(B) which is the fraction of applicants who had that SAT score. Unfortunately, although Pr(A) is very easy to obtain, Pr(B) is not often made public by colleges.

Stanford, though, does provide some rough information about Pr(B) on its Web site, so we can use that as an example. In particular, suppose that someone had a critical reading score of 600 or below. The Stanford admissions statistics show that Pr(B) is 20%. In other words, 20% of applicants had a CR score of 600 or below. Stanford also provides Pr(B | A), which is that 5% of the admitted class had this CR score. Finally, we know Pr(A), which is the overall admit rate, or 4.8%. Applying Bayes’ Theorem, therefore shows that the probability of admission, given just this information about the CR score, equals 0.05 * 0.048 / 0.20, or 1.2%. Stanford goes further to give the actual admit rate in this case, or 1%, but we could easily have computed this by using Bayes’ Theorem.

Of course, in practice, admission is not based on a single piece of information. Therefore, what we really want is the probability of admission, given all of the pieces of information, including GPA, essays, letters of recommendation, state of residence, and so on. Bayes’ Theorem could be applied, but only if we either make a simplifying (but clearly false) assumption that each of these pieces of information are conditionally independent, or if we had the joint conditional and joint marginal distributions, which we are not going to obtain.

I’m hoping that the mathematical formalism helps to clarify things.

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@mdphd92, all true until you tip the windmill from institutional hooks with URM and First-Gen having the greatest impact - 44% URM and 18% First-Gen, understanding that there is some overlap between the two groups.

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Actually, the statistics are not at all informative because colleges superscore & report the subscores separately – so while you posited a composite score range (example 1320-1440 for Chicago’s 5th-25th range) – there is no supporting data for the assumption that a composite of 1320 represents the 5th percentile level. Maybe the lower end of the score spectrums simply represents more lopsided candidates – students with math/verbal score combinations such as 660/7460 (composite 1420 rather than your assumed 1320)

And of course, for reasons I have already pointed out, there is a much higher likelihood of an imbalance in individual scores at the lower end of the spectrum.

By identifying strengths and weaknesses across the board, and doing enough research into individual colleges to to figure out which strengths represent added value to each college, and which weaknesses are not deal killers.

The colleges do publish checklists as to which factors are counted heavily for admission; and it doesn’t take all that much digging to figure out which academic, social, musical or athletic strengths might be particularly appealing to some colleges.

That sample score of 1390 tells absolutely nothing about the student, and has no predictive value at all about chances of admission except for less selective schools where a 1390 might be viewed as particularly impressive. (e.g. representing that school’s 80% level)

The 5% level probably is useful to determine when a student has virtually no chance of admission. (On the other hand, although it was many years ago, my daughter was admitted to Chicago with an SAT composite of 1200 on the old SAT – that’s misleading, however, because my D. opted to submit somewhat stronger ACT scores instead,but it is quite possible that had she enrolled, the school might have later received her weaker SAT scores as well as the ACT scores and reported both on the CDS. But that is just one more problem with your reliance on SAT composites alone).

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The reported ACT/SAT scores is far more cloudy than is believed.

If asked, some colleges will be forthright and inform you that they ‘put their best foot forward’ and report superscored tests to the CDS, US News, etc.

http://www.chronicle.com/blogs/headcount/inflated-sat-scores-reveal-elasticity-of-admissions-data/29575

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@Chembiodad Regarding your post #51 about athletes getting a bump, do you consider that your D had a hook through running, or that she did not have a hook because her apps were RD?

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Thank you very much for the refresher course on conditional probability mdphd92. I learned from that. I think I now understand what you and a couple of others have been saying. The statements I made in the beginning of the thread about the probability of acceptance based on particular SAT percentiles were not accurate. I still think SATs can be useful in determining “reasonable reaches” but I am no longer certain what one’s specific SATs should be to consider a school a reasonable reach.

I know that there have been some pretty sophisticated attempts to calculate such probabilities. Years ago I was fairly proficient at using logistic regression to predict the probability of dichotomous outcomes like college admission (more or less as a personal interest, not part of my career). Logistic regression would be the best statistical tool for actually predicting a particular student’s likelihood of admission to a particular college based on several factors simultaneously but the task of getting together all the data would be daunting.

There was a document written by Jeffrey Simonoff in 2017 that alluded to the use of SATs in predicting college admission via logistic regression.
http://people.stern.nyu.edu/jsimonof/classes/2301/pdf/logistic.pdf
In part, he says:
“We wish to model the probability of being admitted to a somewhat
select college as a function of SAT score. There are three distinct kinds of situations:
(1) For SAT scores around 1050, we might believe that each additional point on the
SAT is associated with a fixed (constant) increase in the probability of being
admitted. That is, a linear relationship is reasonable.
(2) However, for SAT score around 1400, this is no longer true. At that point the
probability of being admitted is so high that an additional point on the SAT adds
little to the probability of being admitted. In other words, the probability curve
as a function of SAT levels off for high values of SAT.
(3) The situation is similar for low SAT (say around 600). At that point the probability
of being admitted is so low that one point lower on the SAT subtracts little
from the probability of being admitted. In other words, the probability curve as
a function of SAT levels off for low values of SAT.
These three patterns suggest that the probability curve is likely to have an S–shape.”

This means that increases in the middle SAT range are more likely to actually increase chances of admission. The law of diminishing returns and diminishing losses would seem to apply. I’m not sure if his statement is based on actual data or if it is a hypothetical example.

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Glad you found the refresher helpful. (Sorry for the multiple postings, which I didn’t intend.) Anyway, the usual lesson from Bayes’ Theorem is the significant effect of the prior probability, Pr(A), which most people don’t consider strongly enough. In other words, getting into Stanford these days is just a low-probability event, regardless of your particular statistics. Likewise, to find a reasonable “reach”, the overall admissions rate is going to be important, not just one’s SAT score.

I agree that a regression model would work well, and since admission is a dichotomous event, such models would need to estimate the probability of admission as a continuous variable, which will end up translating to an S curve output. But, every regression model has an error term, and I suspect that admission models will have rather large error terms, meaning that we can only estimate the probability very approximately from a given applicant’s data, with lots of variability among similar applicants. In short, we can come up with a model, but that’s no guarantee that the model works well for a particular individual. In statistics terms, the bias of a model might be small, but its variance could be very large.

From your original question, I think you might be interested in the probit model (similar to a logit model) developed by Avery and Levin in 2009. (Do a Google search for SIEPR discussion paper 08-31.) Their interest was to explore any differences between early action/decision and regular decision, but in order to do this, they had to first come up with a probability model for admission.

The interesting numbers come up in Table 4. Each column represents a different college, which unfortunately are identified by numbers, rather than names. It shows that a 100 point increase in SAT verbal score, in the mean range of those scores, increased probability of admission by 15 to 19%. But the biggest factor was applying ED or EA vs RD. The second biggest effect was one’s home state, which also modified admission probability by 16 to 20%. (I suspect that if they had modeled URM or first-generation effects, those would also have had large effects as well, especially in today’s climate.) The other thing I find fascinating about this table is how similar the 7 colleges were in evaluating each of the pieces of data.

I do think there is a threshold effect for SAT scores, not merely from the S curve from the logistic or probit regression model, but from the nature of human decision making. I think that human beings tend to use categories as a key heuristic in making decisions, so they will categorize an applicant’s SAT scores into low, medium, and high categories, or something like that, and then also try to categorize other factors, like extracurricular activities and community service, to come up with an approximate response of accept, reject, or discuss further with other committee members. Also, I think that cognitively, admissions officers have different standards of low, medium, or high, based on an applicant’s background.

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I seem to be joining the conversation a bit late, but no one appears to have commented on the fact that at some very good schools (e.g. Wellesley and Smith) the acceptance rate for women is 100% higher than it is for men!

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That doesn’t seem right, since you are dividing by zero somewhere.

If some men applied, then the admission rate for men is 0%, so the ratio of women admission rate to men admission rate involves dividing by zero.

If no men applied, then the admission rate for men involves dividing by zero.