Post Exam Analysis

Its midterm season and that probably means what I think it means: students are getting a cold splash in the face and a huge reality check. Students think that they’re doing pretty well so far: the assignments they’ve handed in have all been good, the lab work has been solid, they are taking concise and coherent notes and they understand everything going on. Everything is going swimmingly, but then they get their midterms back and the mark is less than they expected.


What went wrong?


It happens to the best of us. The midterms are (what I think) the professor’s way of telling students to get on track with all the coursework that has been handed out. It is the professor’s way of telling students that if they do not understand something, they better ask the professor. It is the professor’s way of telling students what they need to know and what they need to learn. However, in light of the professor’s advice, it is up to the students to use the information from the midterm to help them succeed in the final (which usually makes up a larger portion of the overall grade than the midterm does).


Suppose you do badly on a midterm which is worth 30%. The final is worth 70% and you almost ace it, getting 95% on the final. Overall, your grade jumped from 50% on the midterm to 15 / 30 + 66.5 / 70 = 81.5 %. That is quite a substantial leap from a D to an A-. Of course, this doesn’t happen every day, nor does it happen with every student but it tells us something important about midterms and finals. If you do badly on a midterm, you can usually make up for it by having an excellent final.


This point stresses the importance of learning from your midterms and making that jump from only getting 50% on your midterm, to getting 95% on your final. In real world situations, the jump is smaller mostly because students who can get 95% on the final are outstanding students and getting 50% on a midterm means that those students just had a bad day. It is also smaller because students usually do better on their midterms (say 65 – 70% ) and then get 80 – 85% on their final after the jump. Talking about acing your finals is easy, how do we actually plan out this success and make it a reality?


The first thing is to put the midterm into perspective. If you did receive 60% on the midterm, what did you lose 40% of the marks on? Thinking about things one way, 60% may really not be that bad. For example, suppose that you didn’t have any time whatsoever and there was nothing you could do about it. You only had time to study 60% of the material; you then get 60% on the midterm. Although on the outside it means you got a C, it can actually mean that you knew and aced 100% of the material you studied.


Now suppose that you studied 80% of the material and received 60% on the midterm. Again, you receive a C but in actuality you received 60 / 80 = 75%  B+. I’ve definitely been there enough to know that getting 60% on a midterm after only studying 70% of the material covered is quite an accomplishment even if it is only a C on the midterm itself. It has probably happened to you while you were sulking about your horrible grade on the midterm.


How does all this translate to succeeding in the final? Well, getting 100% on a test means not only knowing 100% of the material, but it also means losing 0 marks to things like silly mistakes.


We can analyze the marks lost on the test as follows: (Borrowed from the LEAP website on Post Exam Preparation)


1.

Remembering Formulas / Definitions


To gain these marks on future tests, try using flash cards or mnemonics to help you remember.


2.

Understanding Concepts


Go over your notes and make sure you are writing down clear notes that you can understand. Can you summarize your notes in a few sentences or less? Can you explain concepts learned in class to friends not taking the class?


3.

Knowing how things relate to each other


Use mind maps and concept maps to help you relate things in the notes together. Not only will this be good review, but sometimes visually seeing it will give you a new way of learning things.


4.

Silly Mistakes


Get enough sleep before the exam. This means that you’re not up cramming the night before and that you’ve done plenty of studying beforehand. Do practice exams in real exam environments (losing marks on a practice exam is better than losing marks in the real exam)


Another thing to keep in mind is considering the time that you have to study for the test. Suppose you have a hypothetical test that has 10 questions out of a database of 100 questions. You are given the database of questions and it takes you 5 minutes to read, answer, understand and memorize the answer to one question in the database. You have 400 minutes or about 6 hours and 40 minutes. You now have a choice as to how to allocate your time. You can memorize 80 questions solidly or you can try to memorize all 100 questions (using about 4 minutes each question). Which is the better option?


Memorizing 80 questions solidly means that out of the professor’s 10 questions chosen from the database of 100, you should be able to answer 8 of them solidly (and get an A- overall).


If we evaluate this more closely:


Probability of all 10 questions being from the 80 you memorized = (0.8)^10 = 0.107 = 11%


Probability of 9 questions being from the 80 you memorized = (0.8)^9*(0.2)*10 = 0.268 = 27%


Probability of 8 questions being from the 80 you memorized = (0.8)^8*(0.2)^2*10C2 = 45* (0.8)^8*(0.2)^2 = 0.301 = 30%


Probability of 7 questions being from the 80 you memorized = (0.8)^7*(0.2)^3*10C3 =

120*(0.8)^7*(0.2)^3 = 0.201 = 20%


Probability of 6 questions: 0.08 = 8%

Probability of 5 questions: 252*(0.2)^5*(0.8)^5 = 0.026 = 3%

Probability of 4 questions: 210*(0.2)^6*(0.8)^4 = 0.005 = 0%

Probability of 3 questions: 120*(0.2)^7*(0.8)^3 = 0.00 = 0%

Probability of 2 questions: 45*(0.2)^8*(0.8)^2 = 0.00 = 0%

Probability of 1 question: 10*(0.2)^9*(0.8) = 0.00 = 0 %

Probability of 0 questions: (0.2)^10 = 0.00 = 0 %


And your expected mark is: 0.11*10 + 0.27*9 + 0.30*8 + 0.20*7 + 0.08*6 + 0.03*5 = 7.96

No surprises there since you studied all 80 questions, knew them backwards and forwards and made no silly mistakes on those questions.



What if you studied all 100 questions at 4 minutes each instead of the 5? Let’s also say that there’s a 20% chance on any question that you forget it completely (but you do vaguely recall it on the test).


Again, if we do expected value analysis, the analysis will be the same – BUT, we did not take into account the fact that you only studied for 4 out of the 5 minutes required to fully understand the question and you are prone to silly mistakes on the questions. We also cannot forget that since you did look at all the questions, you may be able to obtain part marks on the questions you couldn’t answer (whereas before you had no idea whatsoever). It’s difficult to know whether this will balance out but all this lengthy and probably quite boring evaluations of the two options tells us some important things:


1.

Use as much time as possible to study for exams (that means starting early – maybe even from day 1)
2.

If you don’t have enough time to study all the material, evaluate how much material you know now. If you plan on getting an A- on the midterm and know about 80% of the material, you should probably be studying the material that you know to make sure you fully understand it and won’t make any silly mistakes on it. If you only know 70% and want that A-, study until you know 80% of the material.


Remember that this analysis is based on a very simple model of a test. It is hard in the real world to evaluate how much material you know, have professors select from a database of 100 questions for your midterms or even know what kind of questions will come up and whether it will fall within the domain of the material you know (probabilities are good for large sample sizes rather than one individual case).