Stanford University

Math Placement

Colorful Blackboard - Placement Exam

The Placement Diagnostic is now open for the 2021-22 academic year.

Placement Diagnostic

The placement diagnostic is required of everyone (regardless of AP Calculus credit) to enroll in the courses listed in the chart below unless you’ve already taken the prerequisite course at Stanford.

The placement diagnostic will recommend the initial math course in which to enroll, and this recommendation is purely advisory. The recommended placement does not itself constitute the fulfillment of any requirements at Stanford (for majors or prerequisites). The diagnostic will help identify areas you may want to review. For additional information regarding the courses recommended in the diagnostic, please see the Introductory Math Courses page.

The Precalculus Refresher in Canvas, consists of videos and associated exercises that are available for use at any time. It goes through the core skills in algebra, graphing, functions, and trigonometry that are necessary for learning and using calculus. It is recommended for anyone who needs to brush up on some of these skills or has background gaps. It is best to do this enough in advance so that you have worked through all relevant parts prior to enrollment in a calculus course here.

Course Placement Diagnostic or Prerequisite Course
Math 19  
Math 20 or Math 19
Math 21 or Math 20
Math 51 or Math 21

Please note: 

The diagnostic gives you the most useful feedback if taken in the quarter before you enroll in your first Math department course at Stanford (if at the level of Math 51 or below). Completing the diagnostic between August 1, 2021 and July 9, 2022 gives you permission to enroll in Math 19, 20, 21, and 51 for the 2021-22 academic year as your first Math department course here. Therefore, if you are planning to not take a first Math department course here at the level of Math 51 or below until the 2022-23 academic year or later then you will need to retake the diagnostic at that point.

Please note that after you have completed a first Math department course here at the level of Math 51 or below, you never need to retake the diagnostic for enrollment in another Math department course.  For example, you can take Math 21 this year and put off Math 51 until next year or later without retaking the diagnostic.

Please also note that the outcome of this diagnostic has no effect on AP calculus credit at Stanford. However, choosing to enroll in a course for which you’ve received Transfer or Advanced Placement credit is subject to the Registrar’s policy on the duplication of credit.

Sample Calculus Course Progressions

The following table lists course plans according to your recommended placement.  If the placement diagnostic recommends that you review specific precalculus topics, we urge you to do this before enrolling.

Recommended Placement




Math 19

Math 19 Math 20 Math 21

Math 20

Math 20 Math 21 Math 51

Math 21

Math 21 Math 51 Math 52 or 53*

Math 51

Math 51 Math 52 or 53* Math 52 or 53*

*Math 52 and 53 are independent of each other, but should be taken after completing Math 51.

Each of Math 20, Math 21, Math 51, Math 52, and Math 53 are offered in every quarter of the academic year.  Math 19 is offered only in the fall and winter quarters.

Frequently Asked Questions

Please plan for approximately 60 to 90 minutes to complete the diagnostic. There is NO time limit, so you may take as long as necessary.

The diagnostic is meant to be taken once and you should plan on completing it in one sitting. However, there is no time limit. Please see the front page of the diagnostic and review the “Basic Information” section for more information. Click here.


You will receive an email within 7 days after you’ve completed the diagnostic indicating your eligibility to enroll. After receiving the email, you can enroll immediately in Axess. *Note: if you are an incoming student, although your prerequisite is now satisfied, you will still have an enrollment hold until you attend the New Student Orientation.

Please see the front page of the diagnostic and review the “Basic Information” section. Click here. 

The diagnostic aims to give an accurate sense of which Math course (19 through 51) would be the most appropriate starting point, based on your knowledge when you take it.

Please see the front page of the diagnostic and review the “Basic Information” section. Click here.

No, the purpose of the diagnostic is to suggest the appropriate first Math course based on your knowledge demonstrated in the diagnostic, but this is purely advisory.

Yes, even if you have AP credit you are still required to take the diagnostic before enrolling in Math 51. Please note, if you do have AP credit and decide to take a calculus course, your units may be repeated.

No, you can enroll directly into any course numbered higher than 51. If you choose to change enrollment from a higher course into Math 51 before the Final Study List deadline you will need to take the diagnostic to complete enrollment.

The Math 50 series addresses real-world applications of the concepts it introduces; it does not involve any theoretical proofs, focusing on conveying an understanding of the principles relevant to all concepts where they may be used. The CME 100 series focuses more on engineering applications with the use of programming. The two course sequences teach material in very different orders; e.g., most of the linear algebra content of Math 51 is not covered in CME 100. The best preparation for taking Math 104 (Applied Linear Algebra), Math 52, or Math 53 is to take Math 51.

It’s best to consider your future coursework and concentration plans. Both CS 229 and CS 230 specifically recommend Math 51 (or courses that rest on Math 51) for their math background; Math 51 is the only course at Stanford whose syllabus covers nearly all of the linear algebra and “matrix calculus” material used in CS 229 and CS 230. The Math 50-series provides multivariable calculus and linear algebra background that is relevant to students in all quantitative majors, both in the School of Humanities and Sciences as well as in the School of Engineering. The Mathematical and Computational Sciences (data science), Chemistry, and Physics majors accept only the Math 50 series to fulfill their math requirement, and the Engineering Physics major strongly recommends the Math 50-series for students who are likely to choose any of the Physics course options for depth. The Math 50 series is also strongly recommended if you may need to take more advanced courses in the Math or Statistics departments in the future.

No, since CME is housed under the ICME program at the engineering school, all further questions about regarding CMEcourses must be directed to ICME.  

Please see the front page of the diagnostic and review the “Basic Information” section. Click here.