Exams, Pass, mark 100! Options

[jim palmer]

Pass grade 8, awsome!
I think I was wrong when I said it was unlikely that a candidate
would score exactly 100. Recalling my rusty maths, the number
of candidates getting each mark would fit a "bell curve".
Suppose 10% fail, 40% get passes, 40% merits, 10% distinctions.
The average mark would be 120, and 80% would get from 100 to
135.
So if 1000 candidates take the exam there would be 800 getting
100 to 135, roughly 23 for each mark.
So a mark of 100 is just as likely as any other, its not the examiner
giving you another mark or two if you narrowly failed.

[Splog]

You are right, approx 40% of candidates do pass, and if there is an even spread across the pass range (100-119) then approx 2% of candidates will score 100.(In practice it would be slightly less.) In 2009, around 270,000 candidates sat the exams, so around 5,400 of them would have scored 100.

[steinberger]

Pass grade 8, awsome!
I think I was wrong when I said it was unlikely that a candidate
would score exactly 100. Recalling my rusty maths, the number
of candidates getting each mark would fit a "bell curve".
Suppose 10% fail, 40% get passes, 40% merits, 10% distinctions.
The average mark would be 120, and 80% would get from 100 to
135.
So if 1000 candidates take the exam there would be 800 getting
100 to 135, roughly 23 for each mark.
So a mark of 100 is just as likely as any other, its not the examiner
giving you another mark or two if you narrowly failed.

Keep going with your exams Jim. They demand a lot of work, and playing the same pieces over and over again can get boring, but they are worth it. (IMG:style_emoticons/default/smile.gif)

[jim palmer]

Thanks, I'm a bit lazy, its so easy to listen to good players rather than
practice hard and perform passably yourself!
I'm listening to Alexandra Dariescu playing Chopin just now. I'll make the effort
when the CD has finished and listen to Olga Goija on her viola. Then I will be inspired
to do some practice myself!

[kenm]

Recalling my rusty maths, the number
of candidates getting each mark would fit a "bell curve".
Suppose 10% fail, 40% get passes, 40% merits, 10% distinctions.
The average mark would be 120, and 80% would get from 100 to
135.

Strictly, the best you could say is "approximate to a bell curve", since the normal distribution extends from -infinity to +infinity and the marks are constrained to the range 1 - 140. However, the more worrying approximation is that 100 - 119 apply to 40% of the candidates and 120 - 129 to another 40%, so the density of merits is twice that of passes.

The distribution of marks is rather awkward to fit to any of the common distributions, since the minimum mark is 0, but has probably never been given. If I had the time, I would try a binomial distribution (p+q)^n, where p+q = 1 and n is large enough to cover the range from the lowest plausible mark up to 140. One then has two parameters to fit this distribution to three measurements (passes, merits and distinctions), so the problem is over-determined, and there will be no perfect answer.

[owainsutton]

The distribution of marks is rather awkward to fit to any of the common distributions, since the minimum mark is 0, but has probably never been given.

According to the assessment criteria, the minimum mark for an exam where all sections are attempted is 50. I'll let you explain how that affects the maths! (IMG:style_emoticons/default/smile.gif) (Also, I'm not sure why you're treating the top mark as 140?)

[Splog]

The bell curve in this case will be skewed to the right. Music exams will be different from other exams with a random population, as the vast majority of people pass, certainly at the lower grades. I guess this is because people aren't put in for exams until it is thought that they will do well. So the peak of the curve won't be half way between the lowest and highest marks. I would imagine it lies somewhere within the high pass/low merit area.

[jim palmer]

No of candidates.......x...x
.............................x...........x
...........................x................x 1000 mixed ability
.........................x....................x
........................x................ .....x
.......................x..........A.............x
.....................x............................x
..................oo...............................x
................o.....o...............................x
.............o.....B.....o 150 low ability.........x
___o______________o_________________x__ mark .

I think quite a good fit for the mark distribution could be obtained
by taking the graph of candidates getting each mark for 1000 musical
mixed-ability candidates (A) and subtracting say 150 low ability
candidates (B) who would not take music exams.

[barry-clari]

No of candidates........x...x
.............................x...........x
...........................x................x 1000 mixed ability
.........................x....................x
........................x................ .....x
.......................x..........A.............x
.....................x............................x
..................oo...............................x
................o.....o...............................x
.............o.....B.....o 150 low ability.........x
___o______________o___________________x__ mark .

I think quite a good fit for the mark distribution could be obtained
by taking marks for 1000 musical mixed-ability candidates (a) and
subtracting marks for say 150 low ability candidates (b)who would
not take music exams.

I don't understand that graph (IMG:style_emoticons/default/sad.gif)

[jim palmer]

I've edited my post a bit, hopefully it is clearer.
A is the graph for a large number, say 1000, of unselected
candidates given a music exam. Each x is the number getting a
particular mark.
B is the graph for the candidates in that 1000 who are not much
good at music, I have just guessed 150 but perhaps 300 would be
nearer.
So the number taking music exams, 850, would be A minus B.
If you try it with pencil and paper you get a "bell curve" with the
left-hand tail largely removed, so that very few fail and lots get
passes and merits. That fits the facts!

[kenm]

I'm not sure why you're treating the top mark as 140

Mental aberration.

The thought strikes me that choice of minimum and maximum marks gives one four parameters to fit to three measurements, so it might be possible to improve estimates for the numbers obtaining particular marks in the middle range at the expense of accuracy at the extremes, estimates for the latter being fairly poor inherently to the nature of the exercise.