I hope that you have had a good (if short!) break. I have now finished adjusting and submitting all final marks, and have posted the solutions and marking scheme for the final exam here. (In order to get the marking scheme in red -- otherwise it would be very hard to read! -- I used the TeX --> DVI --> PS --> PDF build sequence, which seems to have led to some degradation in the quality of the typesetting as opposed to the TeX --> PDF sequence I have used for the other documents in this course; if anyone is interested in the original DVI file, it is available here.) This document came about as follows. Before beginning marking papers, I did what I did for the term test and wrote the final as though I were taking it myself, and then marked my subsequent solutions: what I have posted here is essentially just a typeset version of that handwritten script, taking into account the corrections made when marking it, together with a couple other errors discovered while typing it up. Thus it is meant to be indicative of what I feel could be actually written during an exam sitting, rather than a polished version for reference: I hope this explains its roughness at some points.

Some general observations on the final:

-- The average among students registered for the class who wrote the final was 52.58 marks. Out of the original of 125, this amounts to 42%. Initially I determined, rather arbitrarily (because I couldn't think of anything better at the time), to mark out of 115; this would give an average of 45.7%, and marks adjusted this way were used for the first set of course marks posted last week. After spending more time thinking about it, I decided to mark out of 104 instead: this gives an average of 50.6% on the final, and was essentially what was used for the set of amended course marks posted this week. (So if you haven't checked your course mark since last week, I would advise you to check it again as it may have changed.) This left three students sitting within a fraction of a percent of a modifier division: after consideration of all three individual cases I determined to move all three students upwards. Thus the final letter mark everyone received is the same as would have been obtained had I marked the final out of 100 instead of 104.

-- -- The number 104 was used for the following reason. As noted in the exam solutions, many solutions to problems 5, 6, and 7 (especially 5 and 6) lost many marks for assuming the full time-dependent expressions for the solution coefficients, rather than deriving them. I had explicitly indicated in a pre-test announcement that such derivation was necessary, but on the final review sheet this was not necessarily made clear, especially for the heat equation. Thus I decided roughly to make the marks allocated to this matter extra-credit. I also made some marks for problem 7 which I felt were very difficult to actually obtain extra-credit. This still left me to decide between marking out of 106 and 104 (owing to some ambiguity as to what exactly should be made extra-credit) and I ultimately determined to mark out of the lower number.

-- Having made that determination, the final mark was then computed. The quiz mark was computed as follows: an adjustment factor of 1.2 was applied to Quiz 5; then excused missed quizzes were replaced by the average of the remaining quizzes (according with the statement in the syllabus that the weight of such quizzes would be transferred to the remaining ones); then the lowest of the resulting 9 quiz marks was dropped, and the quiz mark was taken to be the average of the remaining 8 ones. (Dropping the lowest quiz before substituting in for the missing quiz marks would amount to dropping too much weight: as an extreme example -- though this would have run up against other provisions in the marking scheme and hence is just for reference -- should someone have had all the weight for the quizzes transferred to the last two, only the higher of the two would count and that student would effectively have been able to drop roughly half of the quizzes instead of just one.) The final mark was then calculated as (Qavg/10)*32 + (TT/56)*28 + (EX/104)*40 + (QB/10)*4 + HW*0.4, where Qavg is the quiz mark just described, TT is the term test mark, EX is the final exam mark, QB is the bonus quiz mark, and HW is the number of homeworks receiving credit. The resulting percentage was rounded (.5 and above rounded up, below .5 rounded down) and that was the number posted as the final course mark. (One student managed to get a 105: that number was reduced to 100.) I believe every final mark was more than 1 percentage point below a modifier division before rounding.

-- The average course mark (before any rounding or truncating) was 63.24%.

-- Before the adjustment to the exam marking scheme described above, the averages on the individual problems were as follows:

-- -- Q1: 66.63%; Q2: 39.45%; Q3: 69.22%; Q4: 38.33%; Q5(a): 42.26%; Q5(b): 38.25%; Q6: 37.38%; Q7: 31.44%; Q8: 38.44%. The questions with the highest average (Q1 and Q3) were very close to pre-term test material (Q2, which was the only problem entirely focussed on pre-term test material, was extraordinarily long, as those of you who tried to carry it through to the end will doubtless attest!). The uniformity of the marks across the other questions (particularly as respects the questions on Fourier transforms) were largely what left me at a loss when trying to determine how to scale the final exam mark initially: I knew that we went through the material on Fourier transforms very quickly, so if the marks on those questions had been appreciably less than the others, I would have been willing to raise them -- but actually they were some of the highest among the non-Laplace's equation problems.

-- The distribution showed clear multimodality (am I using that term correctly, those of you who know more about statistics than I do?): roughly three distinct bell curves were visible, at the low, middle, and upper ends of the overall marks distributions.

That is probably enough (or perhaps too much?) information. Please let me know if you have any comments! (And for those of you who have read this far: apparently only one of you actually noticed my joke on the announcements on the cover sheet of the exam: at any rate only one student commented on it. Can you find it now? I have added a line in brackets to make it more obvious!)

Thanks to all of you for your hard work during this term, it has been good working with you. If I can be of help with anything in the future please do not hesitate to contact me!

Nathan