College Admission Rates

Harvard's 2016-17 acceptance rate was 5.4%. That seems comforting, but then you remember that the students that apply are self-selecting. They all are the brightest in their school and think they'll be the one in twenty that get admitted According to a page of statistics hanging on my high school's wall near the counseling office, about 5% of students with a 1550 SAT get into Harvard. And according to Prep Scholar, only 1% of SAT takers get a 1500 or above.

Perhaps to get a more accurate view of the brightness of students in various colleges, the number of granted acceptances could be compared to total acceptances. According to the National Center for Education Statistics, 3.6 million are expected to graduate high school this year, with 69.8% of them expected to enroll in college immediately (I use the word college for any undergraduate institution). This means 2.5m total acceptances. Comparing various college's number of admits to the total gives this:

Sheet 1

I then realized this system does not work either; a small liberal arts college might have the same number of admits as Harvard but might not be as good. Some colleges simply may not have the resources to accept and educate as many students as others, which would advantage them in this system. Additionally, while lower-tier colleges may accept many people, a varying amount of students will choose a better college or a worse one

Each university's median SAT score of admits might be a good measure. I remember that US News & World Report, a relatively respected website to which many colleges refer for finding their national rank, holistically evaluates colleges based on multiple factors, but their ranking about 90% matches a ranking simply by SAT score, which means SAT score is strongly linked students' admission chances. It seems unusual that a 3-hour multiple choice test could work as an indicator, but the g-factor theory posits that cognitive ability in one task should be similar for many tasks. People with high SATs may also have excellent school grades, extracurricular service, and essay composition skills,

Median SAT score can be gathered (A), as well as percent of SAT takers who achieve that score (B). A college's reported acceptance rate (C) is probably true at the median SAT score, so multiplying B and C together reveals the true tier of American colleges (D).

Sheet 2

*A through C's data are from PrepScholar

**A is the mean of the 25th and 75th percentile SAT scores. Colleges usually report these values and not their median.

Harvard educates the equivalent of the top 7 in 20,000 college students, UMich the top 9 in 1,000, MSU the top 1 in 8, and Wayne the top 1 in 5.

Modeling

Sheet 3

I took Sheet 2, added a few more colleges and their ABCD values, then found 1/C and 1/D. 1/C takes the reciprocal of a college's acceptance rate to show selectivity for the students that apply. 1/D takes the reciprocal of a college's true tier of students to show selectivity with respect to all SAT takers. Sheet 3 gives this chart, with 1/C as the x-axis and 1/D the y:

Google Sheets has a useful trendline producer. The power series regression, shown below, has the highest R-squared value by far, being .16 higher than the closest value:

The regression's function is y=4.26x^2.11. It is hilariously depressing that someone can see this model and think, "You know how hard it seems to get into a top college looking at their reported acceptance rate? Well, it's exponentially harder."

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Thank you for reading,

George Fane

My Spreadsheet:

https://docs.google.com/spreadsheets/d/1qcO4dykOs4c-XkX1bOYQPgnsnRQsGtZzoXBkV4mKeWI/edit?usp=sharing

One of the PrepScholar sites I used for college information: https://www.prepscholar.com/sat/s/colleges/University-of-Michigan-SAT-scores-GPA

Percentiles for SAT scores:

https://blog.prepscholar.com/sat-percentiles-high-precision-2016