Analysis of the UNAM’s entrance exam

The UNAM is Mexico’s biggest and most important university. To enter it students must either take an exam or graduate from a high school run by the UNAM in less than 4 years with a grade point average of at least 70% (although some majors like medicine require 90% for pase directo). The admission exam is given twice a year, in February and June, and any student from any high school with at least a grade point average of 70% can take it. If the student meets the requirements for entering the UNAM, passing the exam guarantees him admission. The exam has 120 questions.

Depending on the student’s choice of major, the admission exam emphasizes one of four basic areas of study. Let’s say you want to study math: in addition to being tested on the topics every high school student is supposed to know, you’ll get a couple of extra questions about integration by parts; if you want to study biology the extra questions will be about the the Krebs cycle; if you want to study philosophy the exam will probably include extra references to the great works of literature; and if you want to study a social science you’ll be asked about the differences between a grand macchiato and a caffè latte.

Click on the chart to visit the interactive version

Apart from UNAM’s main (and most prestigious) campus of Ciudad Universitaria (CU), the university has several satellite campuses in the Mexico City metro area as well as many others across Mexico. In this post I will only analyse those located in Mexico City, and only for the scholarized system (sistema escolarizado) where the students have to actually sit in a classroom (the UNAM also offers remote TV/Internet classes and under an open system). The admission exam to the UNAM is quite competitive and only a small percentage of those who apply actually get in.

Date	Location	Percentage admitted	Applied	Completed Test	Admitted
2011-06	CU	8.6	30,615	27,515	2,626
2011-06	Not CU	9.1	32,215	29,667	2,916
2012-02	CU	5.2	61,262	55,793	3,192
2012-02	Not CU	6.4	56,012	52,287	3,597
2012-06	CU	5.1	30,944	28,084	1,573
2012-06	Not CU	10.1	32,741	30,354	3,299
2013-02	CU	5.4	63,562	59,425	3,424
2013-02	Not CU	6.7	56,348	53,547	3,800
2013-06	CU	6.3	29,872	26,052	1,868
2013-06	Not CU	13.8	33,744	30,403	4,656

This year Harvard sent out 2,023 offers out of 34,295 applications for an admission rate of 5.9%, which is exactly the same overall admission rate as the UNAM (CU) from June 2011 to June 2013 (though, of course, Harvard has much much lower standards in who it admits).

Admissions by major

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Admissions by area

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Biological Sciences and Health


Social Sciences


Humanities and Arts

Highest scoring majors in each area


Changes in median admission scores from exam to exam (first differences)
Median scores each year

(a standard deviation is 17 points)

One big problem with the admission process at the UNAM is that it doesn’t use Gale-Shapley and thus students have an incentive to not reveal their true preferences:

Imagine a student who is deciding between studying ingeniería mecatrónica and ingeniería mecánica, because mechatronics is very hard to get into, and students only get to apply to one major at each exam, he decides to list mechanical engineering as his choice even when there was a chance (albeit not as big) that he would have been admitted to mechatronics.

Click on the chart to visit the interactive version

Another consequence of not using Gale-Shapley is that some students who do list their true preference are rejected in favor of lower scoring students (this is probably a really big deal).

Click on the chart to visit the interactive version

Imagine a student who lists his or her true preference but who would be willing to entertain studying at an alternative campus with a lower admission score (international relations at CU vs international relations at FES Aragón) or a similar major with lower admittance requirements (international relations vs political science for example) who is rejected from his first choice but obtains a test score higher than the minimum requirement for his alternative.
We can see in the above chart how the percentage of students admitted increases at the lower requirement campuses during the June exam, but there is no such increase at the main CU campus. This is probably a consequence of students who had been rejected after the February exam having another go at being admitted at the lower requirement satellite campuses.

The easiest way to remedy this would be to copy the COMIPEMS exam —which the UNAM helped design— and allow students to list more than one major/campus when applying. Obviously there are a lot of variations and complications the UNAM could use to make its admission process better, for example they could start admitting students from different areas than their first choice major if they had really high test scores. I’m sure the applied math and computing international relations students are up to the task of designing an admittance mechanism that satisfies Gale-Shapley.

There’s also a positive correlation between average test scores and the starting salaries of university graduates (of any university in Mexico City, not just the UNAM) in a poll conducted by Reforma in Mexico City.


P.S. I bet the Escuela Nacional de Artes Plásticas is full of rich kids
P.P.S. Visit the companion website full of interactive charts
P.P.P.S. Source code