The Median is the Message

Our first post concerns an error in the 2016 Mathematical Methods Exam 2 (year 12 in Victoria, Australia). It is not close to the silliest mathematics we’ve come across, and not even the silliest error to occur in a Methods exam. Indeed, most Methods exams are riddled with nonsense. For several reasons, however, whacking this particular error is a good way to begin: the error occurs in a recent and important exam; the error is pretty dumb; it took a special effort to make the error; and the subsequent handling of the error demonstrates the fundamental (lack of) character of the Victorian Curriculum and Assessment Authority.

The problem, first pointed out to us by teacher and friend John Kermond, is in Section B of the exam and concerns Question 3(h)(ii). This question relates to a probability distribution with “probability density function”

    \[  f(x) =   \left\{\aligned &\frac{(210-x)e^{\frac{x-210}{20}}}{400} \qquad && 0\leqslant x \leqslant 210,\\ &0 && \text{elsewhere.} \endaligned\right.}\]

Now, anyone with a good nose for calculus is going to be thinking “uh-oh”. It is a fundamental property of a PDF that the total integral (underlying area) should equal 1. But how are all those integrated powers of e going to cancel out? Well, they don’t. What has been defined is only approximately a PDF,  with a total area of 1 - 23/2e^{21/2} \approx 0.9997. (It is easy to calculate the area exactly using integration by parts.)

Below we’ll discuss the absurdity of handing students a non-PDF, but back to the exam question. 3(h)(ii) asks the students to find the median of the “probability distribution”, correct to two decimal places. Since the question makes no sense for a non-PDF, of course the VCAA have shot themself in the foot. However, we can still attempt to make some sense of the question, which is when we discover that the VCAA has also shot themself in the other foot.

The median m of a probability distribution is the half-way point. So, in the integration context here we want the m for which

a)      \phantom{\quad}  \int\limits_0^m f(x)\,{\rm d}x = \dfrac12.

As such, this question was intended to be just another CAS exercise, and so both trivial and pointless: push the button, write down the answer and on to the next question. The problem is, the median can also be determined by the equation

b)     \phantom{\quad}  \int\limits_m^{210} f(x)\,{\rm d}x = \dfrac12,

or by the equation

c)     \phantom{\quad} \int\limits_0^m f(x)\,{\rm d}x = \int\limits_m^{210} f(x)\,{\rm d}x.

And, since our function is only approximately a PDF, these three equations necessarily give three different answers: to the demanded two decimal places the answers are respectively 176.45, 176.43 and 176.44. Doh!

What to make of this? There are two obvious questions.

1. How did the VCAA end up with a PDF which isn’t a PDF?

It would be astonishing if all of the exam’s writers and checkers failed to notice the integral was not 1. It is even more astonishing if all the writers-checkers recognised and were comfortable with a non-PDF. Especially since the VCAA can be notoriously, absurdly fussy about the form and precision of answers (see below).

2. How was the error in 3(h)(ii) not detected?

It should have been routine for this mistake to have been detected and corrected with any decent vetting. Yes, we all make mistakes. Mistakes in very important exams, however, should not be so common, and the VCAA seems to make a habit of it.

OK, so the VCAA stuffed up. It happens. What happened next? That’s where the VCAA’s arrogance and cowardice shine bright for all to see. The one and only sentence in the Examiners’ Report that remotely addresses the error is:

“As [the] function f  is a close approximation of the [???] probability density function, answers to the nearest integer were accepted”. 

The wording is clumsy, and no concession has been made that the best (and uniquely correct) answer is “The question is stuffed up”, but it seems that solutions to all of a), b) and c) above were accepted. The problem, however, isn’t with the grading of the question.

It is perhaps too much to expect an insufferably arrogant VCAA to apologise, to express anything approximating regret for yet another error. But how could the VCAA fail to understand the necessity of a clear and explicit acknowledgement of the error? Apart from demonstrating total gutlessness, it is fundamentally unprofessional. How are students and teachers, especially new teachers, supposed to read the exam question and report? How are students and teachers supposed to approach such questions in the future? Are they still expected to employ the precise definitions that they have learned? Or, are they supposed to now presume that near enough is good enough?

For a pompous finale, the Examiners’ Report follows up by snarking that, in writing the integral for the PDF, “The dx was often missing from students’ working”. One would have thought that the examiners might have dispensed with their finely honed prissiness for that one paragraph. But no. For some clowns it’s never the wrong time to whine about a missing dx.

UPDATE (16 June): In the comments below, Terry Mills has made the excellent point that the prior question on the exam is similarly problematic. 3(h)(i) asks students to calculate the mean of the probability distribution, which would normally be calculated as \int xf(x)\,{\rm d}x. For our non-PDF, however, we should should normalise by dividing by \int f(x)\,{\rm d}x. To the demanded two decimal places, that changes the answer from the Examiners’ Report’s 170.01 to 170.06.

7 Replies to “The Median is the Message”

  1. Thanks for your article, Marty. It articulates many of my own feelings and the feelings of many teachers I have discussed this question with. Such an obvious error on the exam is unforgivable. But it is the Assessors Report that upsets me the most. The comments about this question in the Report are (in my view) irresponsible and mealy-mouthed. Nowhere is there an acknowledgement that an error was made – it reads as though ‘almost a pdf’ was deliberate. The comments are a complete whitewash in my view), no doubt to avoid any potential *ahem* class-action. I’ll tell you how some inexperienced but diligent teachers will interpret these comments:
    “I better give my students a function that is a close approximation to a pdf as practice in case a similar question appears in the future ….”
    It deeply troubles me that no mention is made of students who said that the median does not exist because the function is not a pdf (prefaced with some appropriate calculations – so as to earn the ‘method mark’!) nor is it ever stated in the Report that such an answer is correct and acceptable.

  2. The first part of this question asks students to find the mean value.

    Perhaps this should be calculated as:

    (\int_{0}^{210} xf(x) dx) / (\int_{0}^{210} f(x) dx) .

  3. This has just been drawn to my attention by Marty.

    The function f CAN be turned into a probability density function (pdf) by changing just one symbol. If 0 \le x \le 210 is replaced by -\infty < x \le 210 (or even more simply x \le 210) then it is a pdf – it is the density of 210 – a Gamma(2,20) random variable. So the "best" answer for the median would be 210 – the median of a Gamma(2,20) random variable, giving 176.433, and the "best" answer for the mean doesn't need a calculator – it is 210 – 2 * 20 = 170.

    It would be a good exercise for second year probability to give them the VCAA published f and ask them the simplest way to turn this into a pdf.

    Mind you – it is a pretty silly random variable because the Gamma(2,20) distribution is that of the waiting time for the second arrival in a Poisson process of rate 20 per unit time. Since we get negative answers possible for 210 – this random variable, perhaps the examiners had time travel in mind!

    1. Thanks very much, Tim. This suggests that possibly the examiners grabbed a genuine pdf and deliberately chopped it off. If so, I think it is easy to answer *why* they would have chopped it off: as far as I am aware, the Methods (and Specialist) curriculum includes no treatment of, or even mention of, improper integrals. (The exception is the normal distribution, where the magic buttons do all the work.) Of course, independent of the specific nonsense discussed here, this is a very silly curriculum decision.

      Anyway, if that is indeed the reasoning of how the pseudo-pdf was created, it in no way justifies the decision to do so. As for the examiners employing time travel, yes that would perhaps then be the silliest modelling aspect to have appeared in a Methods exam. But, only perhaps: the competition is pretty fierce.

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